We find $\delta>0$ and $B>0$ such that $U:=Q^m(u_0,\delta) \times \textnormal{}]x_0-\delta,x_0+\delta[ \textnormal{} \subset X_w$, $\vert{c(w,u,x)}\vert<B$ and $\vert{d_j(w,u,x)}\vert<B$ for $j \in \{1,...,s\}$ and every $(u,x) \in U$. By passing to a suitable subcurve of $\gamma$ if necessary we may assume that $\gamma(y) \in Q^m(u_0,\delta) \times \textnormal{}]x_0-\delta/2,x_0+\delta/2[$ for every $y \in \textnormal{}]0,1[$. Similarly as in the proof of Remark 5.27(1) we find a continuous definable function $\tilde{\gamma}_x \textnormal{}: \textnormal{}]0,1[ \textnormal{} \to \mathbb{R}$ such that $\tilde{\gamma}:=(\gamma_u,\tilde{\gamma}_x)$ is a definable curve in $C_w \cap U$ compatible with $C_w$ and for every $l \in \{0,...,r\}$ we have

$$\lim_{y \searrow 0} \vert{\tilde{\gamma}_x(y)-\mu_l(w,\gamma_u(y))}\vert > 0.$$

(1): Let $j \in \{1,...,s\}$. Note that

$$\lim_{y \searrow 0} \vert{\mathcal{Y}(w,\tilde{\gamma}(y))}\vert^{\otimes p_j} \in \mathbb{R} \setminus \{0\}$$

by Corollary 5.31(1). Since

$$b_j(t) \vert{\mathcal{Y}(t,x)}\vert^{\otimes p_j}e^{d_j(t,x)} \in [-1,1]$$

for every $(t,x) \in C$ we obtain $\lim_{y \searrow 0} b_j(w,\gamma_u(y)) \in \mathbb{R}$. This gives the first part.

Suppose that $p_j$ is $\kappa$-negative. Then by Proposition 5.34(2) and Remark 5.15(1) we have

$$\lim_{y \searrow 0} \vert{\mathcal{Y}(w,\gamma(y))}\vert^{\otimes p_j} = \infty$$

and therefore

$$\lim_{y \searrow 0}b_j(w,\gamma_u(y))=0.$$

(2): Note that there is $\rho>1$ such that $V(t,x) \in \textnormal{}]1/\rho,\rho[$ for every $(t,x) \in C$ and that

$$\lim_{y \searrow 0} \vert{\mathcal{Y}(w,\tilde{\gamma}(y))}\vert^{\otimes q} \in \mathbb{R} \setminus \{0\}$$

by Corollary 5.31(1). Since $\lim_{y \searrow 0} f(w,\tilde{\gamma}(y)) \in \mathbb{R}$ we obtain

$$\lim_{y \searrow 0} a(w,\gamma_u(y)) \in \mathbb{R}.$$

This gives the first part.

Suppose that $q$ is $\kappa$-negative. Then by Proposition 5.34(2) and Remark 5.15(1) we have

$$\lim_{y \searrow 0} \vert{\mathcal{Y}(w,\gamma(y))}\vert^{\otimes q} = \infty$$

and therefore $\lim_{y \searrow 0}a(w,\gamma_u(y))=0$ since $\lim_{y \searrow 0} f(w,\gamma_u(y)) \in \mathbb{R}$.

Note that in the globally subanalytic setting in <cit.> on page 149 $\alpha^*$ defines a $\kappa$-soft function and $\omega^*$ a $\kappa$-persistent function which can be verified easily with Proposition 5.44(1). (There we have $r=0$ and therefore $\kappa=0$.)

5.45 Definition

We call $C$ near with respect to $\mu_\kappa$ near if there is a definable curve $\gamma: \textnormal{} ]0,1[ \textnormal{} \to C$ compatible with $C$ such that
$\lim_{y \searrow 0} (\mu_\kappa(\gamma_t(y))-\gamma_x(y))=0$. Otherwise we call $C$ far with respect to $\mu_\kappa$far.

The functions $\mu_j$ for a $1$-logarithmic scale $\mathcal{Y}$ with $\textnormal{sign}(\mathcal{Y}) = (1,-1)$ where $C$ is far with respect to $\mu_0$ and near with respect to $\mu_1$. (Here we have $k^{\textnormal{ch}} = 0$ and $r=1$.)

5.46 Remark

If $C$ is far with respect to $\mu_\kappa$ then every definable function $\pi(C) \to \mathbb{R}_{\geq 0}$ is $\kappa$-persistent and $\kappa$-soft.

So if it is desired to find a $\kappa$-persistent function $M$ and a $\kappa$-soft function $N$ such that $G|_D$ fulfills specific properties (where $G:H \to \mathbb{C}$ is definable and

$$D:=\{(t,z) \in H \mid N(t)<\vert{z-\mu_\kappa(t)}\vert<M(t)\})$$

it is enough to look at cells which are near with respect to $\mu_\kappa$. In the case that $k^{\textnormal{ch}} \geq 0$ and $\kappa=k^{\textnormal{ch}}$ the boundedness condition for $\Theta$ gives the following result which simplifies technical proofs below enormously.

5.47 Proposition

Let $k:=k^{\textnormal{ch}} \geq 0$ and let $C$ be near with respect to $\mu_k$. Then $\Theta_l=0$ for every $l \in \{k+1,...,r\}$.

Let $\gamma:\textnormal{}]0,1[\textnormal{} \to C$ be a definable curve compatible with $C$ and let

$$\lim_{y \searrow 0}(\mu_k(\gamma_t(y))-\gamma_x(y))=0.$$

Then $\lim_{y \searrow 0} y_k(\gamma(y))=0$ by $(II_k)$ in Proposition 5.28. We do an induction on $l \in \{k+1,...,r\}$.

$l=k+1$: Assume $\Theta_{k+1} \neq 0$. Then there is $\epsilon_{k+1} \in \textnormal{}]0,1[$ such that

$$\vert{\log(\sigma_ky_k(t,x))-\Theta_{k+1}(t)}\vert < \epsilon_{k+1}\vert{\log(\sigma_ky_k(t,x))}\vert$$

for all $(t,x) \in C$. We obtain

$$\lim\limits_{y \searrow 0} \Bigl|1-\frac{\Theta_{k+1}(\gamma_t(y))}{\log(\sigma_ky_k(\gamma(y)))}\Bigl| \leq \epsilon_{k+1}$$

and therefore $\lim_{y \searrow 0} \vert{\Theta_{k+1}(\gamma_t(y))}\vert=\infty$, a contradiction to $(I_{k+1})$ in Proposition 5.28.

$l \to l+1$: Note that $\sigma_{k+1}=-1$ and $\sigma_{k+2}=...=\sigma_l=1$. So we obtain

on $C$ by the inductive hypothesis. Assume $\Theta_{l+1} \neq 0$. Then there is $\epsilon_{l+1} \in \textnormal{}]0,1[$ such that

$$\vert{\log_{l-k}(-\log(\sigma_ky_k(t,x)))-\Theta_{l+1}(t)}\vert < \epsilon_{l+1}\vert{\log_{l-k}(-\log(\sigma_ky_k(t,x)))}\vert$$

for all $(t,x) \in C$. We obtain

$$\lim\limits_{y \searrow 0} \Bigl|1-\frac{\Theta_{l+1}(\gamma_t(y))}{\log_{l-k}(-\log(\sigma_ky_k(\gamma(y))))}\Bigl| \leq \epsilon_{l+1}$$

and therefore $\lim_{y \searrow 0} \vert{\Theta_{l+1}(\gamma_t(y))}\vert=\infty$, a contradiction to $(I_{l+1})$ in Proposition 5.28.

Regular Functions

In this paragraph we define regular functions on $\pi(C)$ as compositions of log-analytic functions and exponentials of functions which fulfill a special boundedness condition depending on $X$. (For example the center $\Theta$ of a logarithmic scale on $C$ fulfills this condition.)

As above we give the definitions in the parametric setting.

5.48 Definition

Let $g:\pi(C) \to \mathbb{R}$ be definable. We call $g$ $C$-consistent in $u$ with respect to $X$consistent if the following holds. Let $w \in \pi_l(C)$. Then for every definable curve $\gamma: \textnormal{}]0,1[\textnormal{} \to C_w$ compatible with $C_w$ with $\lim_{y \searrow 0} \gamma(y) \in X_w$ we have

$$\lim_{y \searrow 0} g(w,\gamma_u(y)) \in \mathbb{R}.$$

So a $C$-consistent function $g:\pi(C) \to \mathbb{R}$ in $u$ with respect to $X$ fulfills the following: If for $(w_0,u_0) \in \pi(X)$ there is an open neighbourhood $U$ of $u_0$ in $\pi(X)_{w_0}$ and $\epsilon,c_1,c_2>0$ such that $L_C(w_0,u)>\epsilon$, $\inf(C_{(w_0,u)})<c_1$ and $\sup(C_{(w_0,u)})>c_2$ for every $u \in \pi(C)_{w_0} \cap U$ then $g_{w_0}$ is bounded at $u_0$, i.e. there is $V \subset U$ open with $u_0 \in V$ such that $g_{w_0}|_{\pi(C)_{w_0} \cap V}$ is bounded. If $C = X$ then $g$ is locally bounded in $u$.

Let $l=0$. A $C$-consistent function $f:\pi(C) \to \mathbb{R}$ in $u$ with respect to $X$ is bounded at $u_0$, but not necessarily bounded at $u_1$ respectively $u_2$.

5.49 Example

Let $l=0$, $m=1$, $C_1:= \textnormal{}]0,1[\textnormal{} \times \textnormal{}]0,1[$ and

$$C_2:=\{(u,x) \in \textnormal{} ]0,1[ \times \mathbb{R} \mid 0<x<u\}.$$

Then $]0,1[ \textnormal{}\to \mathbb{R}, u \mapsto 1/u,$ is $C_1$-consistent in $u$ with respect to $\mathbb{R}_{>0} \times \mathbb{R}$, but not $C_1$-consistent in $u$ with respect to $\mathbb{R}^2$. Note that this function is $C_2$-consistent with respect to any open $X \subset \mathbb{R}^2$ with $C_2 \subset X$.

In the following we give some elementary properties of $C$-consistent functions in $u$ with respect to $X$.

5.50 Remark

Let $Y \subset \mathbb{R}^n \times \mathbb{R}$ be definable with $X \subset Y$ such that $Y_w$ is open for every $w \in \mathbb{R}^l$. A function $g:\pi(C) \to \mathbb{R}$ which is $C$-consistent in $u$ with respect to $Y$ is also $C$-consistent in $u$ with respect to $X$.

5.51 Example

(1) The functions $\Theta_0,...,\Theta_r$ are $C$-consistent in $u$ with respect to $\mathbb{R}^n \times \mathbb{R}$.

(2) Let $l \in \{0,...,\kappa\}$. The function $\pi(C) \mapsto \mathbb{R}, t \mapsto \mu_{j,l}(t),$ is $C$-consistent in $u$ with respect to $\mathbb{R}^n \times \mathbb{R}$ for $j \in \{0,...,l\}$.

(3) The logarithm of a $C$-heir is $C$-consistent in $u$ with respect to $\mathbb{R}^n \times \mathbb{R}$.

(1): Follows from $(I_j)$ in Proposition 5.28 for $j \in \{0,...,r\}$.

(2): Follows from Corollary 5.29(1).

(3): Let $g:\pi(C) \to \mathbb{R}_{>0}$ be a $C$-heir. Note that there is $l \in \mathbb{N}$, an $l$-logarithmic scale $\hat{\mathcal{Y}}$ with center $(\hat{\Theta}_0,...,\hat{\Theta}_l)$ such that $g=\exp(\hat{\Theta}_j)$ for $j \in \{1,...,l\}$. So this property follows from $(I_j)$ in Proposition 5.28 applied to $\hat{\mathcal{Y}}$ for $j \in \{1,...,l\}$.

5.52 Remark

(1) The set of $C$-consistent functions in $u$ with respect to $X$ is an $\mathbb{R}$-algebra with respect to pointwise addition and multiplication.

(2) The set of non-negative $C$-consistent functions in $u$ with respect to $X$ is a divisible monoid with respect to pointwise multiplication.

(3) Let $q \in \mathbb{Q}$. Let $f:\pi(C) \to \mathbb{R}$ be $C$-consistent in $u$ with respect to $X$. Then $\exp(q \cdot f)$ is $C$-consistent in $u$ with respect to $X$.

(4) Let $g:\pi(C) \to \mathbb{R}_{>0}$ be $C$-consistent in $u$ with respect to $X$. Then $\pi(C) \to \mathbb {R}_{>0}, t \mapsto 1/g(t)$, is $\kappa$-persistent (where $\kappa=r$ or $\kappa \in \{k^{\textnormal{ch}},r\}$ if $k^{\textnormal{ch}} \geq 0$).

A very important observation is that most of the data of an $(m+1,X)$-restricted log-exp-analytically prepared function is $C$-consistent in $u$ with respect to $X$ (since the logarithms of the exponentials in such a preparation are locally bounded in $(u,x)$ with respect to $X$ (compare with Definition 4.10 and Proposition 4.12 above)).

5.53 Remark

Let $e \in \mathbb{N}_0 \cup \{-1\}$ and $r \in \mathbb{N}_0$. Let $f:C \to \mathbb{R}, (w,u,x) \mapsto f(w,u,x),$ be $(m+1,X)$-restricted log-exp-analytically $(e,r)$-prepared in $x$ with preparing tuple

with $b=(b_1,...,b_s)$, $\exp(d)=(\exp(d_1),...,\exp(d_s))$ and $P=(p_1,...,p_s)^t$. Then the following holds.

(1) Let $j \in \{1,...,s\}$. The function $b_j$ is $C$-consistent in $u$ with respect to $X$.

(2) Assume that $f$ is locally bounded in $(u,x)$ with reference set $X$. The function $a$ is $C$-consistent in $u$ with respect to $X$.

Compare with the proof of Proposition 5.44. $\blacksquare$

5.54 Definition

A function $f:\pi(C) \to \mathbb{R}$ is called $C$-regular in $u$ with respect to $X$regular if there is a set $E$ of positive definable functions on $\pi(C)$ such that the following holds. The set $\log(E)$ consists only of $C$-consistent functions in $u$ with respect to $X$ and $f$ can be constructed from $E$.

Throughout Section 5.1.3 we say ”consistent” respectively ”regular”  instead of ”$C$-consistent in $u$ with respect to $X$” respectively ”$C$-regular in $u$ with respect to $X$” since $C$ and $X$ are fixed.

With Remark 3.4 we obtain the following.

5.55 Remark

Let $f:\pi(C) \to \mathbb{R}$ be regular and consistent. Then $\exp(f)$ is regular and consistent.

Let $f_1,...,f_m:\pi(C) \to \mathbb{R}$ be regular. Let $G:\mathbb{R}^m \to \mathbb{R}$ be log-analytic. Then $G(f_1,...,f_m)$ is regular.

Let $E$ be a set of positive definable functions on $\pi(C)$ such that $\log(E)$ consists only of consistent functions and $f$ can be constructed from $E$.

(1): With Remark 5.52(3) we see that $\exp(f)$ is consistent. With Remark 3.4(1) we see that $f$ can be constructed from $E \cup \{\exp(f)\}$ and is therefore regular.

(2): Follows with Remark 3.4(2). $\blacksquare$

Since the logarithmic scale which occurs in a prepared restricted log-exp-analytic function has a $C$-nice center, we assume for the rest of Section 5.1 that $\Theta$ is $C$-nice and therefore also regular.

5.56 Example

(1) Let $l \in \{0,...,r\}$. The function $\pi(C) \to \mathbb{R}, t \mapsto \mu_{j,l}(t),$ is $C$-regular and $C$-consistent for $j \in \{0,...,l\}$.

(2) The function $\pi(C) \to \mathbb{R}, t \mapsto \prod_{j=0}^{\kappa-1}e^{q_j \mu_{j,\kappa}(t)},$ is $C$-regular and $C$-consistent for $q_0,...,q_{\kappa-1} \in \mathbb{Q}$.

(3) Suppose that $X \subset \mathbb{R}^2$ is open with $0 \in X$. Let $C:= \textnormal{}]0,1[^2$ and let $h: \textnormal{}]0,1[\textnormal{} \to \mathbb{R}, u \mapsto e^{-1/u}$. Then $h$ is not $C$-regular.

(1): By Remark 5.55(1), Remark 5.52(1) and an easy induction on $j \in \{0,...,l\}$ we see that $\mu_{j,l}$ is $C$-regular and $C$-consistent.

(2): With (1) and Remark 5.55(1) we obtain that $e^{\mu_{j,l}}$ is $C$-regular. With Remark 5.55(2) we obtain that $\prod_{j=0}^{\kappa-1}e^{q_j \mu_{j,\kappa}(t)}$ is $C$-regular. (By considering the gobally subanalytic function $\mathbb{R}^l \to \mathbb{R}, (\alpha_1,...,\alpha_l) \mapsto \alpha_1^{q_1} \cdot ... \cdot \alpha_l^{q_l}$.)
Note that $\exp(\mu_{j,l})$ is $C$-consistent. Therefore $\prod_{j=0}^{\kappa-1}e^{q_j \mu_{j,\kappa}(t)}$ is $C$-consistent (by Remark 5.52(1)/(3)).

(3): Note that $\pi(C) = \textnormal{}]0,1[$. Suppose the contrary. Let $E$ be a set of $C$-regular functions with respect to $X$ such that $h$ can be constructed from $E$. Note that for every $g \in \log(E)$ there is an open neighbourhood $U$ of $0$ in $\pi(X)$ such that $g|_{U \cap ]0,1[}$ is bounded. Therefore $h \sim \beta$ for a log-analytic $\beta: \textnormal{} ]0,1[ \textnormal{} \to \mathbb{R}$ at zero. But we have $h \sim e^{-1/u}$ at zero, a contradiction. $\blacksquare$

Integrability of Logarithmic Scales

We close this subsection by establishing some definability results of parameterized integrals of logarithmic scales in the complex setting. With the preparation result in Section 5.1.4 one gets then easily the desired result on integration for definable holomorphic extensions of prepared restricted log-exp-analytic functions in one variable. In Section 5.2 we use such results to construct the unary high parametric global complexification.

5.57 Notation

Let $s$ and $\tau$ range over $\mathbb{R}$. Let $\pi^+:\mathbb{R}^n \times \mathbb{R} \times \mathbb{C} \to \mathbb{R}^n \times \mathbb{R}, (t,s,z) \mapsto (t,s),$ be the projection on the first $n+1$ real coordinates$\pi^+$.

For a definable function $g:Y \to \mathbb{C}, (t,s,\tau) \mapsto g(t,s,\tau),$ where $Y \subset \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}$ is definable we say that $g$ is bounded in $\tau$bounded in $\tau$ if $g_{(t,s)}$ is bounded on $Y_{(t,s)}=\{\tau \in \mathbb{R} \mid (t,s,\tau) \in Y\}$ for every $(t,s) \in \pi^+(Y)$.

5.58 Convention

For $(t,s) \in \pi(C) \times \mathbb{R}_{>0}$ we parameterize the circle $\partial B(\mu_{\kappa}(t),s) \subset \mathbb{C}$ with the counterclockwise oriented curve

$$\rho:[-\pi,\pi] \to \mathbb{R}, \tau \mapsto \mu_\kappa(t)+(\prod_{j=0}^{\kappa}\sigma_j)s e^{i \tau}.$$

5.59 Definition

For definable functions $N,M:\pi(C) \to \mathbb{R}_{\geq 0}$ and a definable function $G:H \to \mathbb{C}$ we set

$$\mathcal{B}(G,M):=\{(t,z) \in H \mid \vert{G(t,z)}\vert < M(t)\}$$

$$\mathcal{A}(G,N,M):=\{(t,z) \in H \mid N(t) < \vert{G(t,z)}\vert < M(t)\}.$$

5.60 Remark

Let $N,M:\pi(C) \to \mathbb{R}_{\geq 0}$ be definable. For a definable $G:H \to \mathbb{C}$ the sets $\mathcal{B}(G,M) \subset \mathbb{R}^n \times \mathbb{C}$ and $\mathcal{A}(G,N,M) \subset \mathbb{R}^n \times \mathbb{C}$ are definable. We have

$$\mathcal{B}(\mathcal{P}_{0,\kappa},M)=\{(t,z) \in H \mid 0 < \vert{z-\mu_\kappa(t)}\vert < M(t)\}$$

$$\mathcal{A}(\mathcal{P}_{0,\kappa},N,M)=\{(t,z) \in H \mid N(t) < \vert{z-\mu_\kappa(t)}\vert < M(t)\}.$$

5.61 Definition

Let $D \subset \pi(C) \times \mathbb{C}$ be definable. We set$D(\mu_\kappa)$

$$D(\mu_\kappa):=\bigl\{(t,s,\tau) \in \pi(C) \times \mathbb{R}_{>0} \times [-\pi,\pi] \mid \mu_{\kappa}(t)+(\prod_{j=0}^{\kappa}\sigma_j)se^{i \tau} \in D_t \bigl\}.$$

For the rest of Section 5.1.3 let $k:=k^{\textnormal{ch}}$.

5.62 Remark

Let $M,N:\pi(C) \to \mathbb{R}_{\geq 0}$ be definable functions. Suppose that $M<\vert{\mu_r-\mu_k}\vert$ if $k \geq 0$. Let $D:=\mathcal{A}(\mathcal{P}_{0,\kappa},N,M)$. Then

$$D(\mu_\kappa) =\{(t,s) \in \pi(C) \times \mathbb{R}_{>0} \mid N(t)<s<M(t)\} \textnormal{}\times \textnormal{} ]-\pi,\pi[.$$

$$B:=\{(t,s) \in \pi(C) \times \mathbb{R}_{>0} \mid N(t)<s<M(t)\} \textnormal{}\times \textnormal{} ]-\pi,\pi[.$$

Suppose $\kappa=r$. Let $\prod_{j=0}^r \sigma_j=1$. The case ”$\prod_{j=0}^r \sigma_j=-1$” is treated completely similar. For  $t \in \pi(C)$, $s \in \mathbb{R}_{>0}$ and $\tau \in \textnormal{} [-\pi,\pi]$ we set $\delta(t,s,\tau):=\mu_r(t)+ se^{i \tau}$. Note that we have
$\delta(t,s,\tau) \in \mathbb{C}^-_{\mu_r(t)}$ if $k < 0$ respectively $\delta(t,s,\tau) \in \mathbb{C}^-_{\mu_r(t)} \cap \mathbb{C}^+_{\mu_k(t)}$ if $k \geq 0$ for $(t,s,\tau) \in B$. Remark 5.13 gives $B \subset D(\mu_\kappa)$. Let $(t,s,\tau) \in D(\mu_\kappa)$. Then $t \in \pi(C)$, $N(t)<s<M(t)$ and since $\delta(t,s,\tau) \in D_t$ we have $\tau \in \textnormal ]-\pi,\pi[$. (If $\tau \in \{-\pi,\pi\}$ we obtain $\delta(t,s,\tau)=\mu_r(t)-s<\mu_r(t)$, i.e. $\delta(t,s,\tau) \notin \mathbb{C}^-_{\mu_r}$ and therefore $\delta(t,s,\tau) \notin D_t$ by Remark 5.13.) We obtain $(t,s,\tau) \in B$. The case ”$\kappa=k$” is treatened completely similar. $\blacksquare$

5.63 Proposition

Let $M,N:\pi(C) \to \mathbb{R}_{\geq 0}$ be definable functions. Suppose that $M<\vert{\mu_r-\mu_k}\vert$ if $k \geq 0$. Let $D:=\mathcal{A}(\mathcal{P}_{0,\kappa},N,M)$. Let $F:D \to \mathbb{C}$ be definable such that

$$F^*:D(\mu_\kappa) \to \mathbb{C}, (t,s,\tau) \mapsto F(t,\mu_\kappa(t)+\bigl(\prod_{j=0}^\kappa \sigma_j \bigl) se^{i\tau}),$$

is bounded in $\tau$. Suppose that $F^*$ is globally subanalytic in $\tau$ with support function $g:\pi^+(D(\mu_\kappa)) \to \mathbb{R}^l$ for an $l \in \mathbb{N}_0$. Let

$$\Delta:=\{(t,s,z) \in \pi(C) \times \mathbb{R}_{>0} \times \mathbb{C} \mid N(t) < s < M(t), z \in B(\mu_{\kappa}(t),s)\}.$$

Then the function

$$\Psi:\Delta \to \mathbb{C}, (t,s,z) \mapsto \frac{1}{2 \pi i}\int_{\partial B(\mu_{\kappa}(t),s)}\frac{F(t,\xi)}{\xi-z} d\xi,$$

is well-defined and constructible in $z$ with support function

$$h:\pi^+(\Delta) \to \mathbb{R}^l \times \mathbb{R} \times \mathbb{R}, (t,s) \mapsto (g(t,s),\mu_\kappa(t),s).$$

Let $G:\mathbb{R}^l \times \mathbb{R} \to \mathbb{R}$ be globally subanalytic such that for $(t,s,\tau) \in D(\mu_\kappa)$

$$F^*(t,s,\tau) = G(g(t,s),\tau).$$

For $(t,s,z) \in \Delta$ we have

\begin{eqnarray*}
\int_{\partial B(\mu_{\kappa}(t),s)}\frac{F(t,\xi)}{\xi-z} d\xi&=&\int_{-\pi}^{\pi}\frac{F^*(t,s,\tau)}{\mu_{\kappa}(t)+(\prod_{j=0}^{\kappa}\sigma_j)se^{i \tau}-z} i(\prod_{j=0}^{\kappa}\sigma_j)se^{i\tau} d\tau\\
&=&\int_{-\pi}^{\pi}\frac{G(g(t,s),\tau)}{\mu_{\kappa}(t)+(\prod_{j=0}^{\kappa}\sigma_j)se^{i \tau}-z} i(\prod_{j=0}^{\kappa}\sigma_j)se^{i\tau} d\tau.
\end{eqnarray*}

Note that

$$\Delta \times \textnormal{}]-\pi,\pi[ \textnormal{} \to \mathbb{C}, (t,s,z,\tau) \mapsto \frac{(\prod_{j=0}^{\kappa}\sigma_j)se^{i\tau}}{\mu_{\kappa}(t)+(\prod_{j=0}^{\kappa}\sigma_j)se^{i \tau}-z},$$

is bounded in $\tau$. So $\Psi$ is well-defined.

Let $v:=(v_1,...,v_l)$ range over $\mathbb{R}^l$ and $y$ over $\mathbb{R}$. Set

$$Q:=\{(v,x,y,z,\tau) \in \mathbb{R}^l \times \mathbb{R} \times \mathbb{R}_{>0} \times \mathbb{C} \times  \textnormal{}]-\pi,\pi[ \textnormal{} \mid \vert{x-z}\vert<y\}$$

$$\Omega:\mathbb{R}^l \times \mathbb{R} \times \mathbb{R} \times \mathbb{C} \times \mathbb{R} \to \mathbb{C}, (v,x,y,z,\tau) \mapsto $$

$$\left\{\begin{array}{ll} \frac{G(v,\tau)}{x+(\prod_{j=0}^{\kappa}\sigma_j)ye^{i \tau}-z} i (\prod_{j=0}^{\kappa} \sigma_j)ye^{i \tau}, & (v,x,y,z,\tau) \in Q,\\
0, & \textnormal{else.} \end{array}\right.$$

Then $\Omega$ is globally subanalytic since we have that

$$]-\pi,\pi[ \textnormal{} \to \mathbb{C}, \tau \mapsto e^{i \tau}=\cos(\tau)+i\sin(\tau),$$

is globally subanalytic.

Then for $(t,s,z,\tau) \in \Delta \times \textnormal{} ]-\pi,\pi[$ we have $\vert{\mu_\kappa(t)-z}\vert<s$ and therefore $(h(t,s),z,\tau) \in Q$. So we obtain

$$\frac{G(g(t,s),\tau)}{\mu_{\kappa}(t)+(\prod_{j=0}^{\kappa}\sigma_j)se^{i \tau}-z} i(\prod_{j=0}^\kappa \sigma_j) se^{i\tau}=\Omega(h(t,s),z,\tau)$$

for every $(t,s,z,\tau) \in \Delta \times \textnormal{} ]-\pi,\pi[$. By Remark 1.26 there is a constructible $G:\mathbb{R}^l \times \mathbb{R} \times \mathbb{R} \times \mathbb{C} \to \mathbb{C}$ such that $\Psi(t,s,z)=G(h(t,s),z)$ for $(t,s,z) \in \Delta$. So we see that $\Psi$ is constructible in $z$ with support function $h$. $\blacksquare$

Note the following: For $(t,s) \in \pi^+(\Delta)$ the parameterized integral $\Psi$ coincides with $F_t$ by Cauchy's integral formula if $F_t$ can be extended to a holomorphic function on $B(\mu_\kappa(t),\tilde{s})$ where $s<\tilde{s}<M(t)$.

In our treatment below the initial function $F$ is a holomorphic extension of a prepared restricted log-exp-analytic function in one variable. (In the globally subanalytic setting $F$ is the holomorphic extension of a prepared globally subanalytic function in one variable (see Proposition 2.8 in <cit.>). There we have $\kappa=0$ and $\mu_\kappa=\theta$.) Outgoing from Proposition 5.63 we obtain the following

For a definable holomorphic extension $F:\Gamma \to \mathbb{C}$ of a prepared restricted log-exp-analytic function in one variable which we will construct in Section 5.1.4 we have to find a $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ and a $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $F^*|_{\mathcal{A}(\mathcal{P}_{0,\kappa},N,M)}$ is bounded and globally subanalytic in $\tau$. In the construction of the unary high parametric global complexification in Section 5.2 below the parameter $s$ will be ”replaced” by a function $\alpha:\pi(C) \to \mathbb{R}_{\geq 0}$ which arises in a natural way from $N$ and $M$. For further considerations it is necessary that the function $\Psi(t,\alpha(t),z)$ which we obtain after this replacement is constructible in $z$ with regular support function (according to the preparation result in Section 5.1.4). This is the case if $M$ and $N$ are regular and $F^*$ is free-regular in $s$.

5.64 Definition

Let $D \subset \pi(C) \times \mathbb{C}$ be definable. We call a function $f: D(\mu_\kappa) \to \mathbb{C}$ free-regular in $s$free-regular if there is $l \in \mathbb{N}_0$ such that $f$ is globally subanalytic in $\tau$ with support function $g:=(g_1,...,g_l):\pi^+(D(\mu_\kappa)) \to \mathbb{R}^l$ and the following holds for $g$.
There is $p \in \mathbb{N}_0$, a regular function $\beta: \pi(C) \to \mathbb{R}^p$ and for every $j \in \{1,...,l\}$ a log-analytic $h_j:\mathbb{R}^{p+1} \to \mathbb{R}$ such that $g_j(t,s)=h_j(\beta(t),s)$ for every $(t,s) \in \pi^+(D(\mu_\kappa))$.

5.65 Remark

Let $D \subset \pi(C) \times \mathbb{C}$ be definable.

(1) Let $f: D(\mu_\kappa) \to \mathbb{C}$ be globally subanalytic. Then $f$ is free-regular in $s$.

(2) Let $f: D(\mu_\kappa) \to \mathbb{C}$ be globally subanalytic in $\tau$ with log-analytic support function. Then $f$ is free-regular in $s$.

5.66 Remark

Let $D \subset \pi(C) \times \mathbb{C}$ be definable. Let $f: D(\mu_\kappa) \to \mathbb{C}$ be free-regular in $s$. The following holds.

(1) $f$ is globally subanalytic in $\tau$ with support function $g:\pi^+(D(\mu_\kappa)) \to \mathbb{R}^l$ which can be constructed from a set $E$ of positive definable functions such that every $h \in \log(E)$ is consistent.

(2) Let $q \in \mathbb{N}$ and let $G:\mathbb{C}^q \to \mathbb{C}$ be globally subanalytic. Let $J_1,...,J_q:D(\mu_\kappa) \to \mathbb{C}$ be functions which are free-regular in $s$. Then $H:=G(J_1,...,J_q):D(\mu_\kappa) \to \mathbb{C}$ is free-regular in $s$.

(1): Let $g:=(g_1,...,g_l)$. Definition 5.64, Definition 5.54 and Remark 5.55(2) show immediately that $g_j$ can be constructed from a set $E$ of positive definable functions such that every $h \in \log(E)$ is consistent for $j \in \{1,...,l\}$.

(2): Follows directly with Remark 1.25(2). $\blacksquare$

In the rest of the section we focus on the case that our initial function $F$ in Proposition 5.63 above is $(\sigma\mathcal{Z})^{\otimes q}$ for $q \in \mathbb{Q}^{r+1}$. The first step here is to find a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that
$$z_l^*:\mathcal{A}(\mathcal{P}_{0,\kappa},N,M)(\mu_\kappa) \to \mathbb{C}, (t,s,\tau) \mapsto z_j(t,\mu_\kappa(t)+(\prod_{j=0}^\kappa \sigma_j)se^{i\tau}),$$

is free-regular in $s$ for $l \in \{0,...,\kappa\}$. (We will see below that we can choose $N=0$.) This can be done with the following technical result which is also needed to establish the desired preparation result in Section 5.1.4.

5.67 Proposition

Let $U:\pi(C) \to \mathbb{R}_{\geq 0}$ be regular and $\kappa$-persistent. Then the following holds.

(1) Let $0<\delta \leq 1/2$. There is a regular $\kappa$-persistent $\zeta:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $K:=\mathcal{B}(\mathcal{P}_{0,\kappa},\zeta) \subset \mathcal{B}(\mathcal{P}_{l,\kappa},U)$ for $l \in \{0,...,\kappa\}$ and

$$\Bigl|\frac{\mathcal{P}_{j-1,\kappa}(t,z)}{e^{\mu_{\kappa-j,\kappa}(t)}}\Bigl| < \delta$$

for every $(t,z) \in K$ and every $j \in \{1,...,\kappa\}$. If $U>0$ then $\zeta>0$.

(2) Let $W:\pi(C) \to \mathbb{R}_{\geq 0}$ be regular and $\kappa$-soft. Then there is a regular $\kappa$-persistent $M :\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \mathcal{A}(z_\kappa,W,U)$.

We start with the following claim.

Let $0<\delta \leq 1/2$. There are globally subanalytic functions $\lambda_1,\lambda_2: \mathbb{R}_{\geq 0} \to \textnormal{} [0,\delta]$ such that the following holds.

(i) $\lambda_j$ is monotone increasing for $j \in \{1,2\}$.

(ii) For $r \in \mathbb{R}_{\geq 0}$ and $z \in \mathbb{C}$ we have $\vert{\log(1+z)}\vert<r$ if $\vert{z}\vert < \lambda_1(r)$.

(iii) For $r \in \mathbb{R}_{\geq 0}$ and $z \in \mathbb{C}$ we have $r<\vert{\log(1+z)}\vert$ if $\lambda_2(r) < \vert{z}\vert < \delta$ for $z \in \mathbb{C}$.

(iv) We have $\lambda_2(0)=0$, $\lim_{r \searrow 0} \lambda_2(r)=0$ and $\lim_{r \nearrow r_0} \lambda_j(r)>0$ for $r_0 \in \mathbb{R}_{>0}$ and $j \in \{1,2\}$.

Proof of the claim

$$\lambda_1:\mathbb{R}_{\geq 0} \to \textnormal{}[0,\delta], r \mapsto$$

$$\sup\{c \in \textnormal{}[0,\delta[ \textnormal{} \mid \textnormal{for every $z \in \mathbb{C}$ with } \vert{z}\vert < c \textnormal{ it holds } \vert{\log^*(1+z)}\vert < r\}.$$

Note that $\lambda_1$ is well-defined, monotone increasing and we have $\lambda_1(0)=0$. For $r \in \mathbb{R}_{\geq 0}$ set

$$s(r):= \inf\{c \in \textnormal{}[0,\delta[ \textnormal{} \mid \textnormal{for every $z \in \mathbb{C}$ with } c<\vert{z}\vert < \delta \textnormal{ it holds } r < \vert{\log^*(1+z)}\vert\}.$$

Note that $s(r_1) \leq s(r_2)$ for $0 \leq r_1 \leq r_2$. Set

$$\lambda_2:\mathbb{R}_{\geq 0} \to [0,\delta], r \to \left\{\begin{array}{ll} s(r),& s(r) \neq \infty, \\
\delta,&\textnormal{else.} \end{array}\right.$$

Note that $\lambda_j$ is globally subanalytic since $\log^*$ is globally subanalytic and fulfills the desired properties for $j \in \{1,2\}$.

(1): Let $\lambda_1$ be as in the claim. Let $l \in \{0,...,\kappa\}$. Let $\zeta_{l,l}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto U(t)$. Define by descending induction on $j \in \{1,...,l\}$

$$\zeta_{j-1,l}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \lambda_1(\zeta_{j,l}(t)) e^{\mu_{\kappa-j,\kappa}(t)}.$$

By an easy descending induction on $j \in \{0,...,l\}$ one sees with the claim that $\zeta_{j,l}$ is a well-defined regular $\kappa$-persistent function (compare with Remark 5.42(1) and Example 5.43). If $\vert{\mathcal{P}_{j-1,\kappa}(t,z)}\vert<\zeta_{j-1,l}(t)$ then
$$\Bigl|\frac{\mathcal{P}_{j-1,\kappa}(t,z)}{e^{\mu_{\kappa-j,\kappa}(t)}}\Bigl|<\lambda_1(\zeta_{j,l}(t)) \leq \delta$$

and therefore

$$\vert{\mathcal{P}_{j,\kappa}(t,z)}\vert = \vert{ \log \bigl(1+\frac{\mathcal{P}_{j-1,\kappa}(t,z)}{e^{\mu_{\kappa-j,\kappa}(t)}} \bigl) \vert} < \zeta_{j,l}(t)$$

for $j \in \{1,...,\kappa\}$ and $(t,z) \in H$ by definition of $\lambda_1$. So we take

$$\zeta:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{\zeta_{0,0}(t),...,\zeta_{0,\kappa}(t)\}.$$

(2): Let $0 < \delta \leq 1/2$. Let $\lambda_2$ be as in the claim. By (1) there is a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ such that
$K:=\mathcal{B}(\mathcal{P}_{0,\kappa},M) \subset \mathcal{B}(\mathcal{P}_{\kappa},U) = \mathcal{B}(z_\kappa,U)$ and we have

$$\Bigl|\frac{\mathcal{P}_{l-1,\kappa}(t,z)}{e^{\mu_{\kappa-l,\kappa}(t)}}\Bigl| < \delta$$

for every $(t,z) \in K$ and every $l \in \{1,...,\kappa\}$.
Let $N_\kappa:=W$. Define by descending induction on $l \in \{1,...,\kappa\}$

$$N_{l-1}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \lambda_2(N_l(t)) e^{\mu_{\kappa-l,\kappa}(t)}.$$

By an easy descending induction on $l \in \{0,...,\kappa\}$ one sees that $N_l$ is a well-defined regular $\kappa$-soft function. This gives the following: If $N_{l-1}(t)<\vert{\mathcal{P}_{l-1,\kappa}(t,z)}\vert$ then

$$\lambda_2(N_l(t)) < \Bigl|\frac{\mathcal{P}_{l-1,\kappa}(t,z)}{e^{\mu_{\kappa-l,\kappa}(t)}}\Bigl| < \delta$$

and therefore

$$N_l(t) < \vert{ \log \bigl(1+\frac{\mathcal{P}_{l-1,\kappa}(t,z)}{e^{\mu_{\kappa-l,\kappa}(t)}} \bigl) \vert} = \vert{\mathcal{P}_{l,\kappa}(t,z) \vert}$$

for $l \in \{1,...,\kappa\}$ and $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},M)$ by definition of $\lambda_2$. So set $N:=N_0$. We obtain $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \mathcal{A}(z_\kappa,W,U)$. This finishes the proof of Proposition 5.67. $\blacksquare$

5.68 Corollary

There is a regular $\kappa$-persistent function $M:\pi(C) \to \mathbb{R}_{\geq 0}$ with $M<\vert{\mu_r-\mu_k}\vert$ if $k \geq 0$ such that for every $l \in \{0,...,\kappa\}$ the following holds where $D:=\mathcal{B}(\mathcal{P}_{0,\kappa},M)$.

The function

$$\mathcal{P}_{l,\kappa}^*: D(\mu_\kappa) \to \mathbb{C}, (t,s,\tau) \mapsto \mathcal{P}_{l,\kappa}(t,\mu_\kappa(t)+(\prod_{j=0}^\kappa \sigma_j)se^{i\tau}),$$

is free-regular in $s$.

The function

$$z_l^*:D(\mu_\kappa) \to \mathbb{C}, (t,s,\tau) \mapsto z_l(t,\mu_\kappa(t)+(\prod_{j=0}^\kappa \sigma_j) se^{i\tau}),$$

is free-regular in $s$.

By Proposition 5.67(1) there is a regular $\kappa$-persistent $T:\pi(C) \to \mathbb{R}_{\geq 0}$ such that

$$\Bigl|\frac{\mathcal{P}_{j-1,\kappa}(t,z)}{e^{\mu_{\kappa-j,\kappa}(t)}} \Bigl|<1/2$$

for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},T)$ for every $j \in \{1,...,\kappa\}$. Let $M:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{\vert{\mu_r(t)-\mu_k(t)}\vert,T(t)\}$ and $D:=\mathcal{B}(\mathcal{P}_{0,\kappa},M)$.
Note that a regular function $h:\pi(C) \to \mathbb{R}$ considered as a function on $D(\mu_\kappa)$ is free-regular in $s$.

(1): We do an induction on $l$. Note that $\mathcal{P}_{0,\kappa}^*$ is globally subanalytic, because
$$\mathcal{P}_{0,\kappa}^*(t,s,\tau)=(\prod_{j=0}^\kappa \sigma_j) se^{i \tau}=(\prod_{j=0}^\kappa \sigma_j) s(\cos(\tau) +i\sin(\tau))$$

for $(t,s,\tau) \in D(\mu_\kappa)$. So $\mathcal{P}_{0,\kappa}^*$ is free-regular in $s$ by Remark 5.65(1).

$l-1 \to l:$ For $(t,s,\tau) \in D(\mu_\kappa)$ we obtain

$$\mathcal{P}_{l,\kappa}^*(t,s,\tau)=\log^* \bigl(1+\frac{\mathcal{P}_{l-1,\kappa}^*(t,s,\tau)}{e^{\mu_{\kappa-l,\kappa}(t)}} \bigl).$$

Because $\log^*$ is globally subanalytic we see with Remark 5.66(2) that $\mathcal{P}_{l,\kappa}^*$ is free-regular in $s$ since $e^{\mu_{\kappa-l,\kappa}}$ is regular (and therefore $e^{\mu_{\kappa-l,\kappa}}$ is also free-regular in $s$ considered as a function on $D(\mu_\kappa)$ since coordinate projections are globally subanalytic).

(2): For $(t,s,\tau) \in D(\mu_\kappa)$ we obtain with Proposition 5.18

$$z_l^*(t,s,\tau) = \mathcal{P}_{l,\kappa}^*(t,s,\tau)+\sigma_le^{\mu_{\kappa-l-1,\kappa}(t)}$$

for $l \in \{0,...,\kappa-1\}$ and with Corollary 5.19

$$z_\kappa^*(t,s,\tau) = \mathcal{P}_{\kappa,\kappa}^*(t,s,\tau).$$

So we obtain that $z_l^*$ is free-regular in $s$ for every $l \in \{0,...,\kappa\}$ with (1) if $l=\kappa$ respectively (1) and Remark 5.66(2) if $l \in \{0,...,\kappa-1\}$ since $e^{\mu_{\kappa-l-1,\kappa}}$ is regular.

Now we obtain the desired result on integration for a logarithmic scale in the complex setting if $\kappa=r$.

5.69 Corollary

Let $\kappa=r$. Let $q:=(q_0,...,q_r) \in \mathbb{Q}^{r+1}$. There is a regular $r$-persistent function $M:\pi(C) \to \mathbb{R}_{\geq 0}$ with $M<\vert{\mu_r-\mu_k}\vert$ if $k \geq 0$ such that for $D:=\mathcal{B}(\mathcal{P}_{0,r},M)$ the function

$$\mathcal{Z}_{\mu_r}^{\otimes q}: D(\mu_r) \mapsto \mathbb{C}, (t,s,\tau) \mapsto (\sigma\mathcal{Z})^{\otimes q}(t,\mu_r(t)+(\prod_{j=0}^r\sigma_j)se^{i \tau}),$$

is bounded in $\tau$ and free-regular in $s$.

By Corollary 5.68(2) we find a regular $r$-persistent $T:\pi(C) \to \mathbb{R}_{\geq 0}$ such that

$$z_l^*:\hat{D}(\mu_r) \to \mathbb{C}, (t,s,\tau) \mapsto z_l(t,\mu_r(t)+(\prod_{j=0}^r \sigma_j) se^{i \tau}),$$

is free-regular in $s$ for $l \in \{0,...,r\}$ where $\hat{D}:=\mathcal{B}(\mathcal{P}_{0,r},T)$. Then $\sigma_lz_l^*(t,s,\tau) \in \mathbb{C}^-$ for every $(t,s,\tau) \in \hat{D}(\mu_r)$ and $l \in \{0,...,r\}$ by Definition 5.11 since $\mu_r(t)+(\prod_{j=0}^r \sigma_j) se^{i \tau} \in H_t$ for $(t,s,\tau) \in \hat{D}(\mu_r)$. Let

$$\hat{\mathcal{Z}}_{\mu_r}^{\otimes q}: \hat{D}(\mu_r) \mapsto \mathbb{C}, (t,s,\tau) \mapsto (\sigma\mathcal{Z})^{\otimes q}(t,\mu_r(t)+(\prod_{j=0}^r\sigma_j)se^{i \tau}),$$

$$G:(\mathbb{C}^-)^{r+1} \to \mathbb{C}, (w_0,...,w_r) \mapsto \prod_{j=0}^r (w_j)^{q_j}.$$

Note that $G$ is globally subanalytic. We obtain

$$\hat{\mathcal{Z}}_{\mu_r}^{\otimes q}(t,s,\tau)=G(\sigma_0z_0^*(t,s,\tau),...,\sigma_rz_r^*(t,s,\tau))$$

for every $(t,s,\tau) \in \hat{D}(\mu_r)$. Therefore $\hat{\mathcal{Z}}^{\otimes q}_{\mu_r}$ is free-regular in $s$ by Remark 5.66(2).

$$d:\pi(C) \to \mathbb{R}, t \mapsto \min \{\vert{\mu_r(t)-\mu_j(t)}\vert \mid j \in \{0,...,r-1\} \bigl\}.$$

Note that $d$ is regular (by Example 5.56(1) and Remark 5.55(2)) and $r$-persistent (by Example 5.43 and Remark 5.42(3)). Choose $M:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{T(t),d(t)\}$. Let $D:=\mathcal{B}(\mathcal{P}_{0,r},M)$. The following claim finishes the proof of Corollary 5.69.

The function $\mathcal{Z}_{\mu_r}^{\otimes q}=\hat{\mathcal{Z}}_{\mu_r}^{\otimes q}|_{D(\mu_r)}$ is bounded in $\tau$.

Proof of the claim

Assume the contrary. Then there is $(t,s) \in \pi^+(D(\mu_\kappa))$ and a definable curve $\gamma_\tau: \textnormal{}]0,1[\textnormal{} \to \textnormal{}]-\pi,\pi[$ such that $(t,s,\gamma_\tau(y)) \in D(\mu_\kappa)$ for $y \in \textnormal{}]0,1[$ and

$$\lim_{y \searrow 0} \vert{\mathcal{Z}_{\mu_r}^{\otimes q}(t,s,\gamma_\tau(y))}\vert = \lim_{y \searrow 0} \vert{(\sigma\mathcal{Z})^{\otimes q}(t,\mu_r(t)+(\prod_{j=0}^r \sigma_j)se^{i \gamma_\tau(y)})}\vert = \infty.$$

Note that $0<s<d$. Consider $\gamma: \textnormal{}]0,1[ \textnormal{} \to H, y \mapsto (\gamma_t(y),\gamma_z(y))$ with $\gamma_t(y)=t$ and $\gamma_z(y)=\mu_r(t)+(\prod_{j=0}^r \sigma_j)se^{i \gamma_\tau(y)}$ for $y \in \textnormal{}]0,1[$. Note that this curve is definable and compatible with $C$. Since $0<s<d$ we have

$$\lim_{y \searrow 0} \gamma_z(y) - \mu_l(t) = \lim_{y \searrow 0} \mu_r(t) + (\prod_{j=0}^r \sigma_j)se^{i \gamma_\tau(y)} - \mu_l(t) \neq 0$$

for $l \in \{0,...,r\}$ which is a contradiction to Corollary 5.31(1). $\blacksquare$

Now we investigate the case ”$\kappa=k$”: So for the rest of Section 5.1.3 we suppose $k \geq 0$. It is left to show that we can find a regular $k$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $k$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that
$$z_l^*:\mathcal{A}(\mathcal{P}_{0,k},N,M)(\mu_k) \to \mathbb{C}, (t,s,\tau) \mapsto z_l(t,\mu_k(t)+(\prod_{j=0}^k \sigma_j)se^{i\tau}),$$

is free-regular in $s$ for $l \in \{k,...,r\}$.

We start our considerations with the following three technical results.

5.70 Proposition

Let $D \subset H$ be definable such that $D_t$ is connected and $\mathbb{R} \cap D_t \neq \emptyset$ for
$t \in \pi(D)$. Let $f,g:D \to \mathbb{C}^-$ be functions with the following properties.

$f$ and $g$ are continuous in $z$.

We have $f(t,z) \in \mathbb{R}_{>0}$ if $(t,z) \in D \cap (\pi(C) \times \mathbb{R})$.

$f(t,z)g(t,z) \in \mathbb{C}^-$ for every $(t,z) \in D$.

Then for every $(t,z) \in D$

Suppose that there is $(t,z^*) \in D$ such that

$$\log(f(t,z^*)g(t,z^*)) = \log(f(t,z^*))+\log(g(t,z^*)) \pm 2\pi i$$

(and therefore

$$\vert{\textnormal{arg}(f(t,z^*))+\textnormal{arg}(g(t,z^*))}\vert \geq \pi.)$$

Note that

$$D_t \to \mathbb{R}, z \mapsto \textnormal{arg}(f(t,z)) + \textnormal{arg}(g(t,z)),$$

is continuous and that $\vert{\textnormal{arg}(f(t,z)) + \textnormal{arg}(g(t,z))}\vert < \pi$ if $z \in \mathbb{R}$ for $t \in \pi(C)$. By the intermediate value theorem (by taking a definable curve which connects $z^*$ and a point $a \in D_t \cap \mathbb{R}$) there is $z' \in D_t$ such that

$$\vert{\textnormal{arg}(f(t,z'))+\textnormal{arg}(g(t,z'))}\vert = \pi.$$

This gives $\textnormal{arg}(f(t,z')g(t,z'))=\pi$. But then $f(t,z')g(t,z') \in \mathbb{R}_{< 0}$, a contradiction. $\blacksquare$

5.71 Remark

Let $m \in \mathbb{N}_0$. Let $D:=\{z \in \mathbb{C}^- \mid \vert{z}\vert<1/\exp_m(1)\}$ and

$$L_m:D \to \mathbb{C}, z \mapsto \log_m(-\log(z)).$$

Then $L_m$ is well-defined , definable and holomorphic. We have $\log_m(-\log(z)) \in \mathbb{C}^-$ and $\log_m(-\log(z)) \in \mathbb{R}_{>0}$ if $z \in \mathbb{R}_{>0}$ for $z \in D$.

Let $z \in D$. If $z \notin \mathbb{R}_{>0}$ then

$$\log(z)=\log(\vert{z}\vert)+i\arg(z) \in \mathbb{C} \setminus \mathbb{R}$$

since $\arg(z) \neq 0$ and therefore $\log_m(-\log(z)) \in \mathbb{C}^-$. So assume $z \in \mathbb{R}_{>0}$. We obtain
$\exp_{m-1}(1) < -\log(z)$ and therefore

$$\exp_{m-1-l}(1) < \log_l(-\log(z))$$

for every $l \in \{1,...,m-1\}$ (where $\exp_0:=1$). We obtain $\log_m(-\log(z)) > 0$.

5.72 Proposition

Let $m \in \mathbb{N}$. Let

$$c:=\frac{1}{\exp_m\bigl(2\sqrt{\log^2(3/2)+\pi^2} \bigl)}$$

and $D:= \textnormal{}]0,c[\textnormal{} \times \textnormal{}]-\pi,\pi[$. The function

$$T_l:D \to \mathbb{C}^-, (s,\tau) \mapsto \log_l(-\log(se^{i \tau})),$$

is a well-defined globally subanalytic function in $\tau$ with log-analytic support function for every $l \in \{0,...,m\}$.

Note that $se^{i\tau} \in \mathbb{C}^-$ for every $(s,\tau) \in \mathbb{R}_{>0} \textnormal{} \times \textnormal{}]-\pi,\pi[$. So we obtain well-definability with Remark 5.71. Set

$$S_0:D \to \mathbb{C}, (s,\tau) \mapsto \frac{i \tau}{\log(s)},$$

and inductively for $l \in  \{1,...,m-1\}$

$$S_l:D \to \mathbb{C}, (s,\tau) \mapsto \frac{\log(1+S_{l-1}(s,\tau))}{\log_l(-\log(s))}.$$

Let $l \in \{0,...,m-1\}$. Then $S_l$  is a well-defined globally subanalytic function in $\tau$ with log-analytic support function. For every $(s,\tau) \in D$ we have $\vert{S_l(s,\tau)}\vert<1/2$ and

Proof of the claim

Let $(s,\tau) \in D$. We obtain

$$2 \sqrt{\log^2(3/2)+\pi^2}<\log_l(-\log(s)) \textnormal{ }(*)$$

for every $l \in \{0,...,m-1\}$. We do an induction on $l$. The case $l=0$ is clear.

$l-1 \to l:$ With the inductive hypothesis, $(*)$, and the definition of the complex logarithm we see that $S_l$ is well-defined and that $\vert{S_l(s,\tau)}\vert<1/2$ for every $(s,\tau) \in D$. We have for $(s,\tau) \in D$

With the inductive hypothesis and Remark 1.25(2) it follows that $S_l$ is a globally subanalytic function in $\tau$ with log-analytic support function. Additionally we obtain for $(s,\tau) \in D$ with the inductive hypothesis

\begin{eqnarray*}
\log_l(-\log(s)-i\tau) &=&\log(\log_{l-1}(-\log(s)-i\tau))\\
&=&\log_l(-\log(s)) \bigl(1+\frac{\log(1+S_{l-1}(s,\tau))}{\log_l(-\log(s))} \bigl)\\
\end{eqnarray*}

Note that by the claim

\begin{eqnarray*}
&=&\log(\log_{m-1}(-\log(s))) + \log(1+S_{m-1}(s,\tau))\\
&=&\log_m(-\log(s)) + \log^*(1+S_{m-1}(s,\tau)).\\
\end{eqnarray*}

By the claim and Remark 1.25(2) we obtain immediately that $T_m$ is globally subanalytic in $\tau$ with log-analytic support function. So Proposition 5.72 is proven. $\blacksquare$

5.73 Proposition

Let $k=0$ and $\mathcal{Q}_0:=\mathcal{P}_{0,k}$. There is a regular $0$-persistent function $M:\pi(C) \to \mathbb{R}_{\geq 0}$ with $M<\vert{\mu_r-\mu_0}\vert$ such that for $D:=\mathcal{B}(\mathcal{Q}_0,M)$ and every $l \in \{0,...,r\}$ the function

$$z_l^*:D(\mu_0) \to \mathbb{C}, (t,s,\tau) \mapsto z_l(\mu_0(t)+\sigma_0se^{i\tau}),$$

is globally subanalytic in $\tau$ with log-analytic support function.

We may assume that $C$ is near with respect to $\mu_0$. Otherwise we are done with Remark 5.46 by choosing $M=0$. Note that $\sigma_1=-1$, $\sigma_2=...=\sigma_r=1$ and that $\Theta_1=...=\Theta_r=0$ by Proposition 5.47. By Definition 5.2 we have $\mu_0=\Theta_0$, $\mu_1=\Theta_0+\sigma_0$ and for $j \in \{2,...,r\}$ we have $\mu_j=\Theta_0+\sigma_0e^{-e_{j-1}}$ where $e_0:=0$ and $e_m:=\exp(e_{m-1})$ for $m \in \mathbb{N}$. Note that for $(t,z) \in H$

for $l \in \{2,...,r\}$. Let

$$c:=\frac{1}{\exp_{r-1}(2 \sqrt{\log^2(3/2)+\pi^2})}.$$

Then $c<1/\exp_{r-1}(1)=1/e_r=\vert{\mu_r(t)-\mu_0(t)}\vert$ for $t \in \pi(C)$. Consider $M:=c$. With Remark 5.62 we see that

$$D(\mu_0)=\mathcal{B}(\mathcal{Q}_0,c)(\mu_0)=\pi(C) \textnormal{} \times \textnormal{} ]0,c[ \textnormal{} \times \textnormal{} ]-\pi,\pi[.$$

Note that
$z_0^*(t,s,\tau)=\sigma_0se^{i\tau} = \sigma_0s(\cos(\tau) + i\sin(\tau))$ and $z_1^*(t,s,\tau)=\log(s)+i\tau$ for $(t,s,\tau) \in D(\mu_0)$.
So $z_0^*$ and $z_1^*$ fulfill the desired properties since the restriction of the global sine and cosine function on $]-\pi,\pi[$ are globally subanalytic. We obtain for $l \in \{1,...,r-1\}$

\begin{eqnarray*}
&=&\log_l(-\log(se^{i \tau}))
\end{eqnarray*}

for every $(t,s,\tau) \in D(\mu_0)$. With Proposition 5.72 we see that

$$]0,c[ \textnormal{} \times \textnormal{}]-\pi,\pi[\textnormal{} \to \mathbb{C}, (s,\tau) \mapsto \log_l(-\log(se^{i \tau})),$$

is a well-defined globally subanalytic function in $\tau$ with log-analytic support function for $l \in \{0,...,r-1\}$ and we are done with the proof of Proposition 5.73. $\blacksquare$

5.74 Proposition

Let $\mathcal{Q}_0:=\mathcal{P}_{0,k}$. There is a regular $k$-persistent function $M:\pi(C) \to \mathbb{R}_{\geq 0}$ with $M<\vert{\mu_r-\mu_k}\vert$ such that for $D:=\mathcal{B}(\mathcal{Q}_0,M)$ abd every $l \in \{0,...,r\}$ the function

$$z_l^*:D(\mu_k) \to \mathbb{C}, (t,s,\tau) \mapsto z_l(\mu_k(t)+(\prod_{j=0}^k\sigma_j)se^{i\tau}),$$

is free-regular in $s$.

For $j \in \{0,...,k\}$ we set $\mathcal{Q}_j:=\mathcal{P}_{j,k}$.
We may assume that $C$ is near with respect to $\mu_k$. Otherwise we are done with Remark 5.46 by choosing $M=0$. Note that $\sigma_{k+1}=-1$, $\sigma_{k+2}=...=\sigma_r=1$ and that $\Theta_{k+1}=...=\Theta_r=0$ by Proposition 5.47. So for $(t,z) \in H$ we have with Corollary 5.19

for $l \in \{k+2,...,r\}$.

If $k=0$ we are done with Proposition 5.73. So assume $k > 0$.

Fix $\lambda \in \textnormal{}]0,1/2[$ such that

$$\bigl|{\frac{\log(1+z)-z}{z}}\bigl| < 1/2$$

for $z \in \mathbb{C} \setminus \{0\}$ with $\vert{z}\vert<\lambda$. For $l \in \{1,...,k\}$ set

$$\zeta_l:\pi(C) \to \mathbb{R}_{>0}, t \mapsto
\min \Bigl\{\tfrac{1}{e^{2 \vert{\mu_{k-l,k}(t)}\vert}},\tfrac{1}{\exp_{r-k-1}(2\sqrt{\log^2(3/2)+\pi^2})},$$

$$\lambda e^{\mu_{k-l,k}(t)},\tfrac{e^{\mu_{k-l,k}(t)}}{\exp_{r-k-1}(2 \sqrt{\log^2(3/2)+\pi^2})} \Bigl\}.$$

We see that $\zeta_l$ is regular (by Remark 5.55 and Example 5.56(1)) and $k$-persistent (by Example 5.43 and Remark 5.42(3)) for every $l \in \{1,...,k\}$. For $l \in \{1,...,k\}$ set

$$\Omega_l:=\{(t,z) \in H \mid \vert{\mathcal{Q}_{l-1}(t,z)}\vert<\zeta_l(t)\}.$$

By Proposition 5.67(1) there is a regular $k$-persistent $M_l:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{B}(\mathcal{Q}_0,M_l) \subset \Omega_l$ for $l \in \{1,...,k\}$. Consider

$$M:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{\vert{\mu_r(t)-\mu_k(t)}\vert,M_1(t),...,M_k(t)\}.$$

Then $M$ is $k$-persistent and regular. We have $D:=\mathcal{B}(\mathcal{Q}_0,M) \subset \bigcap_{l=1}^k \Omega_l.$
Clearly $D_t$ is connected and $D_t \cap \mathbb{R} \neq \emptyset$ for $t \in \pi(D)$. Let for $l \in \{1,...,k\}$

$$c_l:D \to \mathbb{C}, (t,z) \mapsto \frac{\sigma_{l-1}\mathcal{Q}_{l-1}(t,z)}{e^{\mu_{k-l,k}(t)}}.$$

Then for every $(t,z) \in D$ and $l \in \{1,...,k\}$ we have

$$\mathcal{Q}_l(t,z) = \log(1 + c_l(t,z)).$$

Let $\mathcal{V} \in \{c,\mathcal{Q}\}$. By Remark 5.21 we have

$$(\prod_{j=l}^k \sigma_j) \mathcal{V}_l(t,z) \in \mathbb{C}^-,$$

$$(\prod_{j=l}^k \sigma_j) \mathcal{V}_l(t,z) \in \mathbb{R}_{>0}$$

if $z \in \mathbb{R}$ for every $(t,z) \in D$ and $l \in \{1,...,k\}$. $\textnormal{ } (+_{\mathcal{V}})$

Note also that

$$\vert{\mathcal{V}_l(t,z)}\vert<\frac{1}{\exp_{r-k-1} \bigl(2\sqrt{\log^2(3/2)+\pi^2} \bigl)}$$

for every $(t,z) \in D$ and $l \in \{1,...,k\}$. $(++_\mathcal{V})$

By shrinking $M$ if necessary we may assume with Corollary 5.68(1) that

$$\mathcal{V}_l^*:D(\mu_k) \to \mathbb{C}, (t,s,\tau) \mapsto \mathcal{V}_l(t,\mu_k(t) + (\prod_{j=0}^k\sigma_j) se^{i \tau}),$$

is free-regular in $s$ for every $l \in \{0,...,k\}$. (Therefore

$$z_l^*:D(\mu_k) \to \mathbb{C}, (t,s,\tau) \mapsto z_l(t,\mu_k(t)+(\prod_{j=0}^k \sigma_j)se^{i\tau}),$$

is also free-regular in $s$.) $\textnormal{ }(+++_\mathcal{V})$

For $l \in \{0,...,k\}$ and $m \in \{0,...,r-k-1\}$ we define

$$R_{m,l}:D \to \mathbb{C}, (t,z) \mapsto \log_m(-\log((\prod_{j=l}^k \sigma_j)\mathcal{Q}_l(t,z))).$$

By Remark 5.71, $(+_\mathcal{Q})$ and $(++_\mathcal{Q})$ we obtain that the function $R_{m,l}$ is well-defined, continuous in $z$, and for $(t,z) \in D$ we have $R_{m,l}(t,z) \in \mathbb{C}^-$ and if $z \in \mathbb{R}$ then $R_{m,l}(t,z) \in \mathbb{R}_{>0}$. $(+_R)$

Note that $\sigma_{k+m+1}z_{k+m+1}=R_{m,k}$ for $m \in \{0,...,r-k-1\}$. So the goal is to show that the corresponding $R_{m,l}^*:D(\mu_k) \to \mathbb{C}$ is free-regular in $s$ by an induction on the natural numbers $l$ and $m$. But the problem is that we have to deal with iterated logarithms. So we have to consider some more auxiliary functions and establish additionally properties for them (compare with Claim 1 to Claim 4).

For $l \in \{1,...,k\}$ we set

$$S_{0,l}:D \to \mathbb{C}, (t,z) \mapsto \mu_{k-l,k}(t),$$

and inductively for $m \in \{1,...,r-k-1\}$

$$S_{m,l}:D \to \mathbb{C}, (t,z) \mapsto \log \Bigl(1 + \frac{S_{m-1,l}(t,z)}{R_{m-1,l-1}(t,z)} \Bigl).$$

Let for $l \in \{1,...,k\}$ and $m \in \{0,...,r-k-1\}$

$$T_{m,l}:D \to \mathbb{C}, (t,z) \mapsto \log_m(-\log((\prod_{j=l}^k \sigma_j)c_l(t,z))).$$

By Remark 5.71, $(+_c)$ and $(++_c)$ we obtain that the function $T_{m,l}$ is well-defined, continuous in $z$, and for $(t,z) \in D$ we have $T_{m,l}(t,z) \in \mathbb{C}^-$ and if $z \in \mathbb{R}$ then $T_{m,l}(t,z) \in \mathbb{R}_{>0}$. $(+_T)$

Let for $l \in \{1,...,k\}$

$$b_l:D \to \mathbb{C}, (t,z) \mapsto \frac{\log(1+c_l(t,z))-c_l(t,z)}{c_l(t,z)},$$

$$U_{0,l}:D \to \mathbb{C}, (t,z) \mapsto -\log(1+b_l(t,z)),$$

and inductively for $m \in \{1,...,r-k-1\}$

$$U_{m,l}:D \to \mathbb{C}, (t,z) \mapsto \log \Bigl(1+\frac{U_{m-1,l}(t,z)}{T_{m-1,l}(t,z)} \Bigl).$$

Note that $\vert{c_l(t,z)}\vert<\lambda$ since $\vert{\mathcal{Q}_{l-1}(t,z)}\vert<\lambda e^{\mu_{k-l,k}(t)}$ for $(t,z) \in D$. So we see that $b_l$ is well-defined, continuous in $z$, $\vert{b_l(t,z)}\vert < 1/2$ by the choice of $\lambda$ and $1+b_l(t,z) \in \mathbb{C}^-$ for $(t,z) \in D$. $(+_b)$

Claim 1

Let $l \in \{1,...,k\}$. The function $S_{m-1,l}$ is well-defined, continuous in $z$ and for $(t,z) \in D$ we have

for every $m \in \{1,...,r-k-1\}$.

Proof of Claim 1

Note that for $(t,z) \in D$

$$\vert{\mathcal{Q}_{l-1}(t,z)}\vert<\min \bigl\{\tfrac{1}{e^{2\vert{\mu_{k-l,k}(t)}\vert}}, \tfrac{1}{\exp_{r-k-1}(2\sqrt{\log(3/2)^2+\pi^2})} \bigl\}. \textnormal{ }(*)$$

We show Claim 1 by induction on $m \in \{1,...,r-k-1\}$. Fix $(t,z) \in D$.

$m=1$: That $S_{0,l}$ is well-defined and continuous in $z$ is clear. We obtain with $(*)$

$$\vert{\mu_{k-l,k}(t)}\vert<-1/2 \log(\vert{\mathcal{Q}_{l-1}(t,z)}\vert)$$

and with the definition of the complex logarithm

$$\vert{\mu_{k-l,k}(t)}\vert<1/2 \vert{\log((\prod_{j=l-1}^k \sigma_j)\mathcal{Q}_{l-1}(t,z))}\vert.$$

This gives

$m-1 \to m:$
We obtain well-definability of $S_{m-1,l}$ by the inductive hypothesis. We get

\begin{eqnarray*}
\vert{S_{m-1,l}(t,z)}\vert&=&\bigl|{\log \bigl(1+\frac{S_{m-2,l}(t,z)}{R_{m-2,l-1}(t,z)} \bigl)} \bigl|\\
&<& \sqrt{\log^2(3/2)+\pi^2}
\end{eqnarray*}

by the inductive hypothesis and definition of the complex logarithm. So we obtain by $(*)$

By the definition of the complex logarithm we obtain that

and we are done since $R_{m-1,l-1}$ is continuous in $z$.
$\blacksquare_{\textnormal{Claim }1}$

By Claim 1 and $(+_R)$ we have that $S_{m,l}$ is well-defined, continuous in $z$ and

$$1+\frac{S_{m-1,l}(t,z)}{R_{m-1,l-1}(t,z)} \in \mathbb{C}^-$$

for $(t,z) \in D$, $l \in \{1,...,k\}$ and $m \in \{1,...,r-k-1\}$. $(+_S)$

Claim 2

Let $l \in \{1,...,k\}$. The function $U_{m-1,l}$ is well-defined, continuous in $z$ and for $(t,z) \in D$ we have

for every $m \in \{1,...,r-k-1\}$.

Proof of Claim 2

Fix $(t,z) \in D$. We have

and by $(+_b)$

$$\vert{U_{0,l}(t,z)}\vert < \sqrt{\log^2(3/2)+\pi^2}. \textnormal{ }(**)$$

We do an induction on $m$.

$m=1$: With $(+_b)$ we obtain that $U_{0,l}$ is well-defined and continuous in $z$. With $(**)$ we obtain

and therefore

This gives

$$\vert{U_{0,l}(t,z)}\vert < -1/2\log(\vert{c_l(t,z)}\vert)$$

and with the definition of the complex logarithm

$$\vert{U_{0,l}(t,z)}\vert<1/2 \vert{\log((\prod_{j=l}^k \sigma_j)c_l(t,z))}\vert.$$

We obtain

$m-1 \to m:$
We obtain well-definability of $U_{m-1,l}$ by the inductive hypothesis. We get

\begin{eqnarray*}
\vert{U_{m-1,l}(t,z)}\vert&=&\bigl|{\log \bigl(1+\frac{U_{m-2,l}(t,z)}{T_{m-2,l}(t,z)} \bigl)}\bigl|\\
& < &\sqrt{\log^2(3/2)+\pi^2}
\end{eqnarray*}

by the inductive hypothesis. This gives with $(**)$

This implies

and with the definition of the complex logarithm we obtain

and are done. (We obtain the continuity of $U_{m-1,l}$ in $z$ since $T_{m-2,l}$ is continuous in $z$.)
$\blacksquare_{\textnormal{Claim }2}$

By Claim 2 we have that $U_{m,l}$ is well-defined, continuous in $z$, and we have

$$1+\frac{U_{m-1,l}(t,z)}{T_{m-1,l}(t,z)} \in \mathbb{C}^-$$

for $(t,z) \in D$, $l \in \{1,...,k\}$ and $m \in \{1,...,r-k-1\}$. $(+_U)$

Claim 3

Let $(t,z) \in D$. Let $l \in \{1,...,k\}$. It holds

for every $m \in \{0,...,r-k-1\}$.

Proof of Claim 3

We do an induction on $m$.

$m=0$: We obtain

\begin{eqnarray*}
T_{0,l}(t,z)&=&-\log((\prod_{j=l}^k \sigma_j)c_l(t,z))\\
&=&-\log((\prod_{j=l-1}^k \sigma_j)\mathcal{Q}_{l-1}(t,z))+\mu_{k-l,k}(t)\\
&=& R_{0,l-1}(t,z)+S_{0,l}(t,z).
\end{eqnarray*}

$m-1 \to m$:
By Proposition 5.70, $(+_R)$, $(+_S)$, $(+_T)$ we obtain with the inductive hypothesis

\begin{eqnarray*}
\log(T_{m-1,l}(t,z))&=&\log(R_{m-1,l-1}(t,z)+S_{m-1,l}(t,z))\\
&=&\log(R_{m-1,l-1}(t,z))+\log \Bigl(1+\frac{S_{m-1,l}(t,z)}{R_{m-1,l-1}(t,z)} \Bigl)
\end{eqnarray*}

and therefore

$\blacksquare_{\textnormal{Claim }3}$

Claim 4

Let $(t,z) \in D$. Let $l \in \{1,...,k\}$. It holds

for every $m \in \{0,...,r-k-1\}$.

Proof of Claim 4

We do an induction on $m$.

$m=0$: We obtain

\begin{eqnarray*}
R_{0,l}(t,z) &=& -\log((\prod_{j=l}^k \sigma_j) \mathcal{Q}_l(t,z))=-\log((\prod_{j=l}^k \sigma_j) \log(1+c_l(t,z)))\\
&=&-\log((\prod_{j=l}^k \sigma_j)(c_l(t,z) + \log(1+c_l(t,z)) -c_l(t,z)))\\
&=&-\log((\prod_{j=l}^k \sigma_j)c_l(t,z)(1 + b_l(t,z))).
\end{eqnarray*}

With $(+_b)$, $(+_c)$, $(+_\mathcal{Q})$ and Proposition 5.70 we obtain

$$-\log((\prod_{j=l}^k \sigma_j)c_l(t,z)(1 + b_l(t,z))) =-\log((\prod_{j=l}^k \sigma_j)c_l(t,z)) -\log(1 + b_l(t,z)).$$

$$R_{0,l} = T_{0,l}(t,z) + U_{0,l}(t,z).$$

$m-1 \to m$: By Proposition 5.70, $(+_R)$, $(+_T)$, $(+_U)$ and the inductive hypothesis we obtain

\begin{eqnarray*}
\log(R_{m-1,l}(t,z)) &=& \log(T_{m-1,l}(t,z)+U_{m-1,l}(t,z))\\
&=&\log(T_{m-1,l}(t,z))+\log \Bigl(1+\frac{U_{m-1,l}(t,z)}{T_{m-1,l}(t,z)} \Bigl)
\end{eqnarray*}

and therefore

\begin{eqnarray*}
R_{m,l}(t,z) = \log(R_{m-1,l}(t,z)) = T_{m,l}(t,z)+U_{m,l}(t,z).
\end{eqnarray*}

$\blacksquare_{\textnormal{Claim }4}$

Now we are able to show that the corresponding $R_{m,l}^*$ is free-regular in $s$: The idea is to show simultaneously that $T_{m,l}^*$ and $R_{m,l}^*$ are free-regular in $s$ for $m \in \{0,...,r-k-1\}$ by doing an induction on $l$ and using Claim 1 to Claim 4.

Claim 5

$(I_l)$ for $l \in \{1,...,k\}$: The function

$$T_{m,l}^*:D(\mu_k) \to \mathbb{C}, (t,s,\tau) \mapsto T_{m,l}(t,\mu_k(t)+(\prod_{j=0}^k \sigma_j)se^{i\tau}),$$

is free-regular in $s$ for every $m \in \{0,...,r-k-1\}$.

$(II_l)$ for $l \in \{0,...,k\}$: The function

$$R_{m,l}^*:D(\mu_k) \to \mathbb{C}, (t,s,\tau) \mapsto R_{m,l}(t,\mu_k(t)+(\prod_{j=0}^k \sigma_j)se^{i\tau}),$$

is free-regular in $s$ for every $m \in \{0,...,r-k-1\}$.

Proof of Claim 5

We start with $(II_0)$. Let $(t,s,\tau) \in D(\mu_k)$. We have

$$\vert{z-\mu_k(t)}\vert = \vert{\mathcal{Q}_0(t,z)}\vert < \tfrac{1}{\exp_{r-k-1}(2\sqrt{\log(3/2)^2+\pi^2})}$$

for all $(t,z) \in D$. Note that

for $m \in \{0,...,r-k-1\}$. Therefore with Proposition 5.72 and Remark 5.65(2) we are done.

Assume that $(I_{l-1})$ and $(II_{l-1})$ have already been shown for $l \in \{1,...,k\}$ where $(I_0)$ is any true statement.
We show $(I_l)$. By Claim 3 we have $T_{m,l}=R_{m,l-1}+S_{m,l}$ on $D$ for every $m \in \{0,...,r-k-1\}$. With $(II_{l-1})$ and Remark 5.66(2) it suffices to show that

$$S_{m,l}^*:D(\mu_k) \to \mathbb{C}, (t,s,\tau) \mapsto S_{m,l}(t,\mu_k(t)+(\prod_{j=0}^k \sigma_j)se^{i\tau}),$$

is free-regular in $s$ for every $m \in \{0,...,r-k-1\}$. We do an induction on $m$. For $m=0$ this is clear since $S_{0,m}^*$ considered as a function on $\pi(D)$ is the restriction of the regular function $\pi(C) \to \mathbb{R}, t \mapsto \mu_{k-l,k}(t)$. So with Definition 5.64 one sees immediately that $S_{0,m}^*$ is free-regular in $s$. Assume $m>0$. By Claim 1 we obtain that

for every $(t,z) \in D$. So

$$S_{m,l}=\log^* \Bigl(1+\frac{S_{m-1,l}}{R_{m-1,l-1}} \Bigl).$$

With $(II_{l-1})$, the inductive hypothesis and Remark 5.66(2) we are done.

We show $(II_l)$. By Claim 4 we have $R_{m,l}=T_{m,l}+U_{m,l}$ on $D$ for every $m \in \{0,...,r-k-1\}$. With $(I_l)$ and Remark 5.66(2) it suffices to show that

$$U_{m,l}^*:D(\mu_k) \to \mathbb{R}, (t,s,\tau) \mapsto U_{m,l}(t,\mu_k(t)+(\prod_{j=0}^k \sigma_j)se^{i\tau}),$$

is free-regular in $s$ for every $m \in \{0,...,r-k-1\}$. We do an induction on $m$.

$m=0$: Note that $U_{0,l}(t,z)=\log^*(1+b_l(t,z))$  and

for every $(t,z) \in D$ (compare with $(+_b)$ and $(++_c)$). With $(+++_c)$ and Remark 5.66(2) we are done.

$m-1 \to m$:
By Claim 4 we obtain that

$$\Bigl|\frac{U_{m-1,l}(t,z)}{T_{m-1,l}(t,z)} \Bigl|<1/2$$

for every $(t,z) \in D$. So

$$U_{m,l}=\log^* \Bigl(1+\frac{U_{m-1,l}}{T_{m-1,l}} \Bigl).$$

With $(I_l)$, the inductive hypothesis and Remark 5.66(2) we are done.
$\blacksquare_{\textnormal{Claim }5}$

We have

for every $m \in \{0,...,r-k-1\}$. Therefore we obtain with $(II_k)$ in Claim 5 that $z_{k+m+1}^*$ is free-regular in $s$ for every $m \in \{0,...,r-k-1\}$. By $(+++_\mathcal{Q})$ we also have that $z_l^*$ is free-regular in $s$ for $l \in \{0,...,k\}$. This finishes the proof of Proposition 5.74.$\blacksquare$

5.75 Corollary

Let $q:=(q_0,...,q_r) \in \mathbb{Q}^{r+1}$. There is a $k$-persistent regular function $M:\pi(C) \to \mathbb{R}_{\geq 0}$ with $M<\vert{\mu_r-\mu_k}\vert$ such that for $D:=\mathcal{B}(\mathcal{P}_{0,k},M)$ the function

$$\mathcal{Z}_{\mu_k}^{\otimes q}: D(\mu_k) \mapsto \mathbb{C}, (t,s,\tau) \mapsto (\sigma\mathcal{Z})^{\otimes q}(t,\mu_k(t)+(\prod_{j=0}^k\sigma_j)se^{i \tau}),$$

is bounded in $\tau$ and free-regular in $s$.

The proof for this corollary is the same as the proof of Corollary 5.69. $\blacksquare$

§.§.§ Holomorphic Extensions of Prepared Restricted

Log-Exp-Analytic Functions in one Variable and the Desired Result on Integration

In this section we compute a definable holomorphic extension $F:\Lambda \to \mathbb{C}$ of a prepared restricted log-exp-analytic function $f:C \to \mathbb{R}$ in one variable and construct for $\kappa = r$ or $\kappa \in \{k^{\textnormal{ch}},r\}$ if $k^{\textnormal{ch}} \geq 0$ a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that the following holds: We have $D:=\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \Gamma$ and $F|_D$ is complex log-analytically prepared in $z$ with regular data, i.e. of the form

$$F|_D=a(t) (\sigma_0z_0)^{q_0} \cdot ... \cdot (\sigma_rz_r)^{q_r} U(t,z)$$

where $a(t)$ is regular, $q_0,...,q_r \in \mathbb{Q}$ and $U=V \circ \phi$ where $V$ is a convergent complex power series and the components of $\phi$ are of the form $b_j(t) (\sigma_0z_0)^{p_0} \cdot ... \cdot (\sigma_rz_r)^{p_r}$ where $b_j(t)$ is regular and $p_0,...,p_r \in \mathbb{Q}$. This fact shows that $F|_D$ depends ”only in a globally subanalytically way” from $z_0|_D,...,z_r|_D$. Finally we obtain the desired result on integration with our considerations in Section 5.1.3.

At first we define complex log-analytically prepared functions.

5.76 Definition

Let $Y \subset H$ be definable. Let $r \in \mathbb{N}_0$. Let $F:Y \to \mathbb{C}$ be a function. We say that $F$ is complex $r$-log-analytically prepared in $z$ with center $\Theta$complex!log-analytically prepared if

$$F(t,z)=a(t) (\sigma\mathcal{Z})^{\otimes q}(t,z) U(t,z)$$

for all $(t,z) \in Y$ where $a$ is a definable function on $\pi(Y)$ which vanishes identically or has no zero, $q \in \mathbb{Q}^{r+1}$, and $U=V \circ \phi$ such that the following holds.

(a) The function $\phi$ is given by

$$\phi:Y \to \overline{D^s(0,1)}, (t,z) \mapsto$$

where $s \in \mathbb{N}_0$, $b_1,...,b_s:\pi(Y) \to \mathbb{R}$ are definable which have no zeros and $p_{1l},...,p_{sl} \in \mathbb{Q}$ for every $l \in \{0,...,r\}$.

(b) $V$ is a power series which converges absolutely on an open neighbourhood of $\overline{D^s(0,1)}$.

We call $a$ coefficientcoefficient and $b_1,...,b_s$ are called base functionsbase functions. We call $\mathcal{LA}_\mathbb{C}:=(r,\mathcal{Z},a,q,s,V,b,P)$ where $\mathcal{Z}:=(z_0,...,z_r)$ and $b:=(b_1,...,b_s)$ and

\cdot&& &\cdot\\
\cdot&& &\cdot\\
\end{array}\right)\in M\big(s\times (r+1),\mathbb{Q})$$

a complex LA-preparing tuplecomplex LA-preparing tuple for $F$.

The next Remark shows that a log-analytically prepared function $f:C \to \mathbb{R}$ in $x$ can be extended to a complex log-analytically prepared function in $z$ which is holomorphic in $z$, i.e. there is a definable $\Gamma \subset \mathbb{R}^n \times \mathbb{C}$ with $C \subset \Gamma$ and a complex $r$-log-analytically prepared function $F:\Gamma \to \mathbb{C}$ in $z$ such that $\Gamma_t$ is open, $F_t$ is holomorphic for $t \in \pi(C)$ and $F|_C=f$.

5.77 Remark

Let $r \in \mathbb{N}_0$. Let $f: C \to \mathbb{R}$ be $r$-log-analytically prepared in $x$. Let $\mathcal{LA}:=(r,\mathcal{Y},a,q,s,v,b,P)$ be a preparing tuple for $f$ with $b:=(b_1,...,b_s)$ and $P:=(p_1,...,p_s)^t$. Let $R>1$ be such that $v$ converges absolutely on an open neighbourhood of $[-R,R]^s$. For $l \in \{1,...,s\}$ set

$$\Gamma_l:=\{(t,z) \in H \mid \vert{b_l(t)(\sigma\mathcal{Z})^{\otimes p_l}(t,z)}\vert < R\}.$$

$$\Gamma:=\bigcap_{l=1}^s \Gamma_l$$

and consider

$$F:\Gamma \to \mathbb{C}, (t,z) \mapsto a(t)(\sigma\mathcal{Z})^{\otimes q}(t,z) \hat{V}(b_1(t) (\sigma\mathcal{Z})^{\otimes p_1}(t,z),...,b_s(t)(\sigma\mathcal{Z})^{\otimes p_s}(t,z))$$

where $\hat{V}$ is a complex power series in $s$ variables which converges absolutely on an open neighbourhood of $\overline{D^s(0,R)}$ with $\hat{V}|_{[-1,1]^s}=v$. Then the function $F$ is well-defined, definable and holomorphic in $z$. Additionally we have $C \subset \Gamma$, $F|_C=f$ and $F$ is complex $r$-log-analytically prepared in $z$ with preparing tuple $\mathcal{LA}_\mathbb{C}:=(r,\mathcal{Z},a,q,s,V,1/R \cdot b,P)$ where $V(z_1,...,z_s):=\hat{V}(Rz_1,....,Rz_s)$.

Now we compute a definable holomorphic extension of a prepared restricted log-exp-analytic function in one variable. At first we consider a function $f:C \to \mathbb{R}$ which is log-exp-analytically $(e,r)$-prepared with respect to a set $E$ of positive definable functions on $C$.

For Section 5.1.4 we fix the following. Let $e \in \mathbb{N}_0 \cup \{-1\}$ and $r \in \mathbb{N}_0$. Fix a function $f:C \to \mathbb{R}$ which is log-exp-analytically $(e,r)$-prepared in $x$ with center $\Theta$ with respect to a finite set $E$ of positive definable functions such that every $g \in \log(E)$ is log-exp-analytically $(l,r)$-prepared in $x$ with respect to $E$ for a $l \in \{-1,...,e-1\}$ (compare with Definition 4.9). For $g \in \log(E) \cup \{f\}$ we fix a corresponding preparing tuple $(r,\mathcal{Y},a_g,\exp(d_{0,g}),q_g,s,v_g,b_g,\exp(d_g),P_g)$ where $b_g:=(b_{1,g},...,b_{s,g})$ and $\exp(d_g):=(\exp(d_{1,g}),...,\exp(d_{s,g}))$ (compare with Definition 4.9). Fix a finite set $\mathcal{E}$ of $C$-heirs such that $\Theta$ and $a_g,b_g$ can be constructed from $\mathcal{E}$ for $g \in \log(E) \cap \{f\}$.

Fix $R > 1$ such that $v_g$ converges absolutely on $[-R,R]^s$ and a complex power series $V_g$ which converges absolutely on $\overline{D^s(0,R)}$ with $V_g|_{[-1,1]^s}=v_g$ for every $g \in \log(E) \cup \{f\}$. By redefining $a_g$ if necessary we may suppose that $\vert{V_g}\vert<\pi$ (since $V_g$ is bounded. This implies also $v_g<\pi$).

5.78 Definition

For a log-exp-analytically $(l,r)$-prepared function $g \in \log(E) \cup \{f\}$ we define by induction on $l \in \{-1,...,e\}$ a set $\Lambda_g \subset \pi(C) \times \mathbb{C}$ $\Lambda_g$ and a function $\Phi_g:\Lambda_g \to \mathbb{C}$ $\Phi_g$ which is complex $(l,r)$-prepared in $z$complex!$(l,r)$-prepared and a complex preparing tuple $\mathcal{J}_{\mathbb{C}}$complex!preparing tuple for $\Phi_g$. To this preparation we assign a set $E_{\Phi_g}$$E_{\Phi_g}$ of complex definable functions on $\Lambda_g$.

Base case: $l=-1$: We set $\Lambda_g=\pi(C) \times \mathbb{C}$, $\Phi_g=0$, $E_{\phi_g}=\emptyset$ and $\mathcal{J}_{\mathbb{C}}=(0)$.

Inductive Step: Let

where $b_g:=(b_1,...,b_s)$, $e^{d_g}:=(e^{d_1},...,e^{d_s})$, $P_g:=(p_1,...,p_s)^t$ and for $d_i$ the corresponding $\Lambda_{d_i}$, the complex log-exp-analytically $(l-1,r)$-prepared $\Phi_{d_i}$ in $z$ and the corresponding set $E_{\phi_{d_i}}$ have already been defined for $i \in \{0,...,s\}$. Set

$$\Lambda_{0,g}:=\{(t,z) \in \Lambda_{d_0} \cap H \mid \vert{\textnormal{Im}(\Phi_{d_0}(t,z))}\vert<\pi\}$$

and for $i \in \{1,...,s\}$ set

$$\Lambda_{i,g}:=\{(t,z) \in \Lambda_{d_i} \cap H \mid \vert{b_i(t)(\sigma\mathcal{Z})^{\otimes p_i}(t,z)\exp(\Phi_{d_i}(t,z))}\vert < R, \vert{\textnormal{Im}(\Phi_{d_i}(t,z))}\vert<\pi\}.$$

Set $\Lambda_g=\bigcap_{i=0}^s \Lambda_{i,g}$ and

$$\Phi_g:\Lambda_g \to \mathbb{C}, (t,z) \mapsto a(t)(\sigma\mathcal{Z})^{\otimes q}(t,z) e^{\Phi_{d_0}(t,z)}$$

$$V_g(b_1(t)(\sigma\mathcal{Z})^{\otimes p_1}(t,z)e^{\Phi_{d_1}(t,z)},...,b_s(t)(\sigma\mathcal{Z})^{\otimes p_s}(t,z)e^{\Phi_{d_s}(t,z)}).$$

We set

$$E_{\Phi_g}=(E_{\Phi_{d_0}} \cup E_{\Phi_{d_1}} \cup ... \cup E_{\Phi_{d_s}} \cup \{e^{\Phi_{d_0}},...,e^{\Phi_{d_s}}\}) |_{\Lambda_g}$$

for $\Phi_g$ where $\exp(\Phi_d):=(\exp(\Phi_{d_1}),...,\exp(\Phi_{d_s}))$.

We often write $F$ instead for $\Phi_g$ and $\Lambda$ instead for $\Lambda_g$ if $g=f$.
$\Lambda_g$$\Phi_g$complex! log-exp-analytically $(l,r)$-preparedcomplex!preparing tuple$E_{\Phi_g}$

5.79 Remark

Let $g \in \log(E) \cup \{f\}$ be log-exp-analytically $(l,r)$-prepared in $x$ with respect to $E$ for $l \in \{-1,...,e\}$. We have that $\Lambda_g$ is definable, $(\Lambda_g)_t$ is open for $t \in \pi(C)$ and $\Phi_g$ is holomorphic in $z$. Additionally we have $\Phi_g|_C=g$ and $\Phi_g$ can be constructed from $E_{\phi_g} \cup \mathcal{E}$.

We show that $\Phi_g$ can be constructed from $E_{\Phi_g} \cup \mathcal{E}$. The other properties are clear. We do an induction on $l$. For $l=-1$ there is nothing to show.

$l-1 \to l:$ Let

where $b_g:=(b_1,...,b_s)$, $\exp(d_g):=(\exp(d_1),...,\exp(d_s))$ and $P_g:=(p_1,...,p_s)^t$. Note that $a$ and $b$ can be constructed from $\mathcal{E}$. Let

be a complex preparing tuple for $\Phi_g$ where $D_j:=\Phi_{d_j}$ for $j \in \{0,...,s\}$ and

Note that $z_0,...,z_r$ can be constructed from $\mathcal{E}$ by Remark 5.12(3) since $\Theta_0,...,\Theta_r$ can be constructed from $\mathcal{E}$. By the inductive hypothesis we have that the functions $D_0,...,D_s$ can be constructed from $E_{D_0} \cup ... \cup E_{D_s} \cup \mathcal{E}$. So we obtain that $D_0|_{\Lambda_g},...,D_s|_{\Lambda_g}$ can be constructed from $(E_{D_0} \cup ... \cup E_{D_s})|_{\Lambda_g} \cup \mathcal{E}$ and we see with Remark 3.10(1) that the functions $e^{D_0}|_{\Lambda_g},...,e^{D_s}|_{\Lambda_g}$ can be constructed from $E_{\phi_g} \cup \mathcal{E}$ since the imaginary part of $D_j$ is bounded and $e^{D_j}|_{\Lambda_g} \in E_{\Phi_g}$ for $j \in \{0,...,s\}$. Let

$$\eta:\Lambda_g \to \mathbb{C} \times \mathbb{C}^s \times \mathbb{C}^{s+1}, (t,z) \mapsto (a(t),b_1(t),...,b_s(t),e^{D_0(t,z)},...,e^{D_s(t,z)}).$$

Then $\eta$ can be constructed from $E_{\Phi_g} \cup \mathcal{E}$.

There is a globally subanalytic $J:\mathbb{C} \times \mathbb{C}^s \times \mathbb{C}^{s+1} \times \mathbb{C}^{r+1} \to \mathbb{C}$ such that

for every $(t,z) \in \Lambda_g$.

Proof of the claim

Let $u:=(u_0,...,u_s)$ and $y:=(y_0,...,y_s)$ range over $\mathbb{C}^{s+1}$. Let $q=:(p_{00},...,p_{0r})$. Let $w:=(w_0,...,w_r)$ range over $\mathbb{C}^{r+1}$. For $l \in \{0,...,s\}$ let

$$\alpha_l:\mathbb{C}^{s+1} \times \mathbb{C}^{s+1} \times \mathbb{C}^{r+1} \to \mathbb{C}, (u,y,w) \mapsto $$

$$\left\{\begin{array}{ll} u_ly_l \prod_{j=0}^r (\sigma_jw_j)^{p_{lj}}, & \sigma_jw_j \in \mathbb{C}^- \textnormal{ for every } j \in \{0,...,r\}, \\
0, & \textnormal{else.} \end{array}\right.$$

$$J:\mathbb{C}^{s+1} \times \mathbb{C}^{s+1} \times \mathbb{C}^{r+1} \to \mathbb{C}, (u,y,w) \mapsto $$

$$\left\{\begin{array}{ll} \alpha_0(u,y,w)V(\alpha_1(u,y,w),...,\alpha_s(u,y,w)), & \vert{\alpha_i(u,y,w)}\vert \leq 1 \textnormal{ for all } i \in \{1,...,s\}, \\
0, & \textnormal{else}. \end{array}\right.$$

Then $J$ is globally subanalytic. Note that $\sigma_lz_l(t,z) \in \mathbb{C}^-$ for every $l \in \{0,...,r\}$ and $(t,z) \in H$. Therefore

for every $(t,z) \in \Lambda_g$.

With Remark 3.10(2) we are done with the proof of Remark 5.79.

Now we consider restricted log-exp-analytically prepared functions and so we set up the parametric setting from Section 5.1.3.

For the rest of this section we set the following. Let $l,m \in \mathbb{N}_0$ be with $n=l+m$. Let $w$ range over $\mathbb{R}^l$, $u$ over $\mathbb{R}^m$ and $x$ over $\mathbb{R}$. Let $X \subset \mathbb{R}^n \times \mathbb{R}$ be definable with $C \subset X$ such that $X_w$ is open for $w \in \mathbb{R}^l$. We suppose that $\log(E)$ consists only of restricted log-exp-analytic functions in $(u,x)$ with reference set $X$ which are locally bounded in $(u,x)$ with reference set $X$. So $f:C \to \mathbb{R}, (w,u,x) \mapsto f(w,u,x),$ is $(m+1,X)$-restricted log-exp-analytically $(e,r)$-prepared in $x$ and every $g \in \log(E)$ is $(m+1,X)$-restricted log-exp-analytically $(l,r)$-prepared in $x$ for an $l \in \{0,...,e-1\}$.

We say ”consistent” respectively ”regular” instead of ”$C$-consistent in $u$ with respect to $X$” respectively ”$C$-regular in $u$ with respect to $X$”. Let $k:=k^{\textnormal{ch}}$. We fix $\kappa=r$ or $\kappa \in \{k,r\}$ if $k \geq 0$.

The expression ”$\kappa$-persistent” respectively ”$\kappa$-soft” means always ”$(C,\mathcal{Y},\kappa)$-persistent in $u$ with respect to $X$” respectively ”$(C,\mathcal{Y},\kappa)$-soft in $u$ with respect to $X$”.

We start our considerations with some examples: It may happen that there is an unbounded $g \in \log(E)$ such that $\Phi_g$ has bounded imaginary part or there can be a bounded $g \in \log(E)$ such that $\Phi_g$ is unbounded and has a bounded imaginary part.

5.80 Example

Let $w=0$. Let $X=C:= \textnormal{}]0,1[^2$. Let $E:=\{C \to \mathbb{R}, (u,x) \mapsto \exp(u \log(x))\}$. Then $E$ is a set of one positive definable function with $\log(E)=\{g\}$ where $g:C \to \mathbb{R}, (u,x) \mapsto u \log(x),$ is locally bounded in $(u,x)$ with reference set $X$, but not bounded. Consider

$$f:C \to \mathbb{R}, (u,x) \mapsto \exp(u \log(x))$$

Then $f$ is $(2,X)$-restricted log-exp-analytically $(1,1)$-prepared in $x$ with center $0$ (since $f$ is $(1,1)$-prepared in $x$ with respect to $E$, the underlying logarithmic scale is $\mathcal{Y}=(x,\log(x))$ and $v_f=v_g=1$). Note that $\mu_0=0$ and $\mu_1=1$. Therefore $H=\textnormal ]0,1[ \times (\mathbb{C}^-_0 \cap \mathbb{C}^+_1)$. We see that $\Lambda=\Lambda_g=H$ (since $\textnormal{Im} (u\log(z))<\pi$ for $(u,z) \in H$), $\Phi_g:\Lambda \to \mathbb{C}, (u,z) \mapsto u \log(z),$ and

$$F:\Lambda \to \mathbb{C}, (u,z) \mapsto \exp(u \log(z)).$$

Note that $\Phi_g$ has bounded imaginary part, but is unbounded at zero. Let

$$D:= \{(u,x) \in \textnormal{}]0,1[ \textnormal{} \times \textnormal{} \mathbb{R}_{>0} \mid e^{-1/u} < x < 1\}$$

and consider $h:=f|_D$. Then $h$ is $(2,X)$-restricted log-exp-analytically $(1,1)$-prepared in $x$ since $h$ is log-exp-analytically $(1,1)$-prepared in $x$ with respect to $E|_D$. Note that $\log(E|_D)=\{g|_D\}$ is a set of one bounded function. We have $\Lambda_h=\Lambda_{g|_D}=\Lambda$ and $\Phi_h=F$ (since $v_h=v_{g|_D}=1$ and therefore $V_h=V_{g|_D}=1$).

In the following we establish the desired preparation result. The proof is very sophisticated and needs some more preparations.

For a definable function $T:\pi(C) \to \mathbb{R}_{\geq 0}$ we set

$$T_{\textnormal{pers}}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \left\{\begin{array}{lll} 1/T(t),& T(t) \neq 0, \\
0,&\textnormal{else,} \end{array}\right.$$

$$T_{\textnormal{soft}}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \left\{\begin{array}{lll} 1/T(t),& T(t) \neq 0, \\
1,&\textnormal{else.} \end{array}\right.$$

Note that $T_{\textnormal{pers}}$ and $T_{\textnormal{soft}}$ are regular if $T$ is regular.$T_{\textnormal{pers}}$$T_{\textnormal{soft}}$

5.81 Proposition

Let $T:\pi(C) \to \mathbb{R}_{\geq 0}$ be a regular $\kappa$-persistent function. Let $q \in \mathbb{Q}^{r+1}$ be with $(q_\kappa,...,q_r) \neq 0$. The following holds.

(1) Let $q_\kappa>0$. Then there is a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{B}(\mathcal{P}_{0,\kappa},M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$.

(2) Let $k \geq 0$. Suppose that $\kappa=k$ and let $q$ be $k$-positive with $q_k=0$. Suppose that $T_{\textnormal{pers}}$ is consistent. Then there is a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{B}(\mathcal{P}_{0,\kappa},M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$.

(3) Suppose that $T_{\textnormal{soft}}$ is $\kappa$-soft and $q_\kappa<0$. Then there is a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T).$

We may assume that $C$ is near with respect to $\mu_\kappa$. Otherwise we are done with Remark 5.46 by choosing $M=0$ in (1) respectively (2) and $N=M=0$ in (3). Note that if $\kappa=k$ then $\sigma_{k+1}=-1$ and $\sigma_{k+2}=...=\sigma_r=1$ and $\Theta_{\kappa+1}=...=\Theta_r=0$ by Proposition 5.47 and therefore by definition of $H$

$$\log_l(1/\sigma_kz_k(t,z)) = \log_{l-1}(-\log(\sigma_kz_k(t,z))) \in \mathbb{C}^-$$

for $(t,z) \in H$ and $l \in \{1,...,r-k\}$. By Proposition 5.18 we have

$$\prod_{j=0}^r \vert{z_j(t,z)}\vert^{q_j} = \prod_{j=0}^{\kappa-1} \vert{\sigma_j\mathcal{P}_{j,\kappa}(t,z)+e^{\mu_{\kappa-j-1,\kappa}(t)}}\vert^{q_j} \prod_{l=\kappa}^r \vert{z_l(t,z)}\vert^{q_l}$$

for $(t,z)\in H$. We have $(t,z) \in \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$ if

$$\prod_{l=\kappa}^r \vert{z_l(t,z)}\vert^{q_l} < T(t) \prod_{j=0}^{\kappa-1} \frac{e^{-q_j\mu_{\kappa-j-1,\kappa}(t)}}{\vert{\frac{\sigma_j\mathcal{P}_{j,\kappa}(t,z)}{e^{\mu_{\kappa-j-1,\kappa}(t)}}+1}\vert^{q_j}}$$

for $(t,z) \in H$. By Proposition 5.67(1) there is a regular $\kappa$-persistent $M^+:\pi(C) \to \mathbb{R}_{\geq 0}$ such that for every $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},M^+)$ and $j \in \{0,...,\kappa-1\}$ we have

$$\bigl|\frac{\sigma_j\mathcal{P}_{j,\kappa}(t,z)}{e^{\mu_{\kappa-j-1,\kappa}(t)}} \bigl|<1/2.$$

So we have $(t,z) \in \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$ if

$$\prod_{l=\kappa}^r \vert{z_l(t,z)}\vert^{q_l} < \rho T(t)\prod_{j=0}^{\kappa-1} e^{-q_j\mu_{\kappa-j-1,\kappa}(t)} \textnormal{ }(+)$$

for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},M^+)$ where $\rho:=\prod_{j=0}^{\kappa-1} (1/3)^{\vert{q_j}\vert}$.

$$\phi:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \rho T(t)\prod_{j=0}^{\kappa-1} e^{-q_j\mu_{\kappa-j-1,\kappa}(t)}.$$

Note that $\phi$ is regular by Example 5.56(1) and Remark 5.55(2) and $\kappa$-persistent by Remark 5.42(1) (compare also with Example 5.43).

To be able to do necessary estimations with the iterated logarithms in $(+)$ and to construct the desired functions $M$ respectively $N$ in the appropriate cases below we need the following Claim 1.

Claim 1

Let $l \in \mathbb{N}$, $m \in \mathbb{N}$ and $z \in \mathbb{C}^-$ with $\exp_l(m^2\pi)<\vert{z}\vert$. Then $\vert{\log_l(z)}\vert < \sqrt[2^l m]{\vert{z}\vert}$.

Proof of Claim 1

For $p \in \mathbb{N}$ we set $c_p:=\exp(p^2\pi)$. We do an induction on $l$.

$l=1$: Assume $c_m<\vert{z}\vert$. We show

$$\sqrt{\vert{z}\vert^{-1/m}(\log^2(\vert{z}\vert)+\arg^2(z))} < 1.$$

One sees that

$$[c_m,\infty[ \textnormal{} \to \mathbb{R}_{\geq 0}, x \mapsto \log^2(x)x^{-1/m},$$

is strictly monotone decreasing and that the function

$$\mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}, x \mapsto x^4\pi^2/\exp(x\pi),$$

takes its values on $]0,1/2[$. Therefore

$$\log^2(c_m)c_m^{-1/m}=\log^2(c_m)/\exp(m\pi) = m^4\pi^2/\exp(m\pi)<1/2$$

and consequently

$$\textnormal{arg}^2(z) \vert{z}\vert^{-1/m}<1/2.$$

$l-1 \to l$: Assume $\exp_l(m^2 \pi) < \vert{z}\vert$. We have by the definition of the complex logarithm

$$c_1 \leq c_m < \log_{l-1}(\vert{z}\vert) \leq \vert{\log_{l-1}(z)}\vert.$$

Therefore by the base case for $m=1$ applied to $\log_{l-1}(z)$

$$\vert{\log_l(z)}\vert  = \vert{\log(\log_{l-1}(z))}\vert < \sqrt{\vert{\log_{l-1}(z)}\vert}$$

and by the inductive hypothesis

$$\sqrt{\vert{\log_{l-1}(z)}\vert} < \sqrt{\sqrt[2^{l-1} m]{\vert{z}\vert}} = \sqrt[2^l m]{\vert{z}\vert}.$$

$\blacksquare_{\textnormal{Claim }1}$

If $k \geq 0$ we set

$$m:=\max\{\lceil{\vert{q_j/q_b}\vert}\rceil \mid j \in \{b+1,...,r\}\}$$

where $b=k$ if $q_k \neq 0$ or $b=l_{\min}(q)$ if $q_k=0$.

(1): Suppose that $q_\kappa>0$. We do two cases.

Case 1:

Assume $\kappa=r$ or if $\kappa=k$ (i.e. $k \geq 0$) then $q_{k+1}=...=q_r=0$. Let

$$\hat{M}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \sqrt[q_\kappa]{\phi(t)}.$$

Then $\hat{M}$ is $\kappa$-persistent by Remark 5.42(2) and regular by Remark 5.55(2). If $\vert{z_\kappa(t,z)}\vert<\hat{M}(t)$ then $\vert{z_\kappa(t,z)}\vert^{q_\kappa} < \phi(t)$ for $(t,z) \in H$ (so the inequality in $(+)$ is fulfilled for such a $(t,z)$). By Proposition 5.67(1) there is a regular $\kappa$-persistent $M^*:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{B}(\mathcal{P}_{0,\kappa},M^*) \subset \mathcal{B}(z_\kappa,\hat{M})$. So take $M:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{M^*(t),M^+(t)\}$. Clearly $M$ is regular by Remark 5.55(2) and $\kappa$-persistent by Remark 5.42(3). With $(+)$ we have $\mathcal{B}(\mathcal{P}_{0,\kappa},M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$.

Case 2:

Assume $\kappa=k$ (i.e. $k \geq 0$), $q_k>0$ and $(q_{k+1},...,q_r) \neq 0$. Set

$$\hat{M}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{\sqrt[s]{\phi(t)},1/\exp_{r-k}(m^2\pi)\},$$

where $s:=\frac{q_k}{2^{r-k}}$. Then $\hat{M}$ is regular and $k$-persistent.

Let $(t,z) \in \mathcal{B}(z_k,\hat{M})$. We show that the estimation in $(+)$ holds. We have

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} \vert{z_k(t,z)}\vert^{-\tfrac{1}{2^j}} < \sqrt[q_k]{\phi(t)}$$

and therefore

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} (1/\vert{z_k(t,z)}\vert)^{\tfrac{1}{2^j}} < \sqrt[q_k]{\phi(t)}.$$

We obtain

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} (1/\vert{z_k(t,z)}\vert)^{\bigl|\tfrac{q_{k+j}}{2^j m q_k}\bigl|} < \sqrt[q_k]{\phi(t)}$$

since $1/\vert{z_k(t,z)}\vert > \exp_{r-k}(m^2\pi)$ and $\vert{\tfrac{q_{k+j}}{m q_k}}\vert \leq 1$. With Claim 1 applied to $1/\sigma_kz_k(t,z)$ we obtain

$$\vert{\log_j(1/\sigma_kz_k(t,z))}\vert < \sqrt[2^jm]{1/\vert{z_k(t,z)}\vert}$$

for $j \in \{1,...,r-k\}$ and therefore

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} \vert{\log_j(1/\sigma_kz_k(t,z))}\vert^{\vert{q_{k+j}/q_k}\vert} < \sqrt[q_k]{\phi(t)}.$$

Because $\vert{\log_j(1/\sigma_kz_k(t,z))}\vert>1$ for every $j \in \{1,...,r-k\}$ we obtain

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} \vert{\log_j(1/\sigma_kz_k(t,z))}\vert^{q_{k+j}/q_k} < \sqrt[q_k]{\phi(t)}.$$

Now define  $M$ outgoing from $\hat{M}$ in the same way as in Case 1. With $(+)$ we have $\mathcal{B}(\mathcal{P}_{0,k},M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$.

(2): Suppose that $\kappa=k$, that $q$ is $k$-positive with $q_k=0$ and that $T_{\textnormal{pers}}$ is consistent. We have that $\phi_{\textnormal{pers}}$ is consistent by Example 5.51(2), Remark 5.52(1)/(3) (since $T_{\textnormal{pers}}$ is consistent). Let $l:=l_{\min}(q)$. Note that $q_l<0$. Set

$$\hat{M}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto $$

$$\left\{\begin{array}{lll} \min \{1/\exp_{l-k}(\sqrt[s]{\phi_{\textnormal{pers}}(t)}),1/\exp_{r-k}(m^2\pi)\} ,& \phi_{\textnormal{pers}}(t) \neq 0, \\
0,&\textnormal{else,} \end{array}\right.$$

where $s:=-(1/2)^{r-l}q_l$. Since $\phi_{\textnormal{pers}}$ is consistent we obtain that $\sqrt[s]{\phi_{\textnormal{pers}}}$ is consistent by Remark 5.52(2). By Remark 5.52(3) we see that $\exp_j(\sqrt[s]{\phi_{\textnormal{pers}}})$ is also consistent for $j \in \{1,...,l-k\}$. So we see that $\hat{M}$ is regular by Remark 5.55(2).

Claim 2

$\hat{M}$ is $\kappa$-persistent.

Proof of Claim 2

Let $\gamma:\textnormal{}]0,1[\textnormal{} \to C_w$ be a definable curve compatible with $C_w$ with
$\lim_{y \searrow 0} \gamma(y) \in X_w$ and

$$\lim_{y \searrow 0} (\gamma_x(y) - \mu_\kappa(w,\gamma_u(y)))=0.$$

Since $\phi$ is $\kappa$-persistent we see $\lim_{y \searrow 0} \phi(w,\gamma_u(y))>0$ and therefore

$$\lim_{y \searrow 0} \phi_{\textnormal{pers}}(w,\gamma_u(y)) \in \mathbb{R}_{>0}.$$

$\blacksquare_{\textnormal{Claim }2}$

Let $(t,z) \in \mathcal{B}(z_k,\hat{M})$. Then $\phi_{\textnormal{pers}}(t) \neq 0$. We show that the inequality in $(+)$ is satisfied for this $(t,z)$. Let $y:=\log_{l-k}(1/\sigma_kz_k(t,z))$. Note that

$$\max\{\sqrt[s]{\phi_{\textnormal{pers}}(t)},\exp_{r-l}(m^2\pi)\} < \vert{y}\vert.$$

We obtain with $\phi_{\textnormal{pers}}(t)=1/\phi(t)$

$$\vert{y}\vert \prod_{j=l+1}^r \vert{y}\vert^{-\tfrac{1}{2^{j-l}}} > \sqrt[q_l]{\phi(t)}$$

and therefore

$$\vert{y}\vert \prod_{j=l+1}^r (1/\vert{y}\vert)^{\tfrac{1}{2^{j-l}}} > \sqrt[q_l]{\phi(t)}.$$

This gives

$$\vert{y}\vert \prod_{j=l+1}^r (1/\vert{y}\vert)^{\bigl|\tfrac{q_j}{2^{j-l} m q_l}\bigl|} > \sqrt[q_l]{\phi(t)}$$

since $\vert{y}\vert>1$ and $\vert{\tfrac{q_j}{m q_l}}\vert \leq 1$. Since $\exp_{r-l}(m^2\pi)<\vert{y}\vert$ we obtain with Claim 1 applied to $y$

$$\vert{\log_{j-l}(y)}\vert < \sqrt[2^{j-l}m]{\vert{y}\vert}$$

for $j \in \{l+1,...,r\}$ and therefore

$$\vert{y}\vert \prod_{j=l+1}^r \vert{\log_{j-l}(y)}\vert^{-\vert{q_j/q_l}\vert} > \sqrt[q_l]{\phi(t)}.$$

Because $\vert{\log_{j-l}(y)}\vert>1$ for every $j \in \{l+1,...,r\}$ we obtain

$$\prod_{j=l}^r \vert{\log_{j-k}(1/\sigma_kz_k(t,z))}\vert^{q_j/q_l} > \sqrt[q_l]{\phi(t)}$$

and therefore

$$\prod_{j=l}^r \vert{\log_{j-k}(1/\sigma_kz_k(t,z))}\vert^{q_j} < \phi(t).$$

Now define $M$ outgoing from $\hat{M}$ in the same way as in $(1)$ in Case 1. With $(+)$ we have $\mathcal{B}(\mathcal{P}_{0,k},M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$.

(3): Suppose that $T_{\textnormal{soft}}$ is $\kappa$-soft and $q_\kappa<0$. We see with Remark 5.52(3) that $\phi_{\textnormal{soft}}$ is also $\kappa$-soft. We have to consider two more cases.

Case 1:

Assume $\kappa=r$ or if $\kappa=k$ then $q_{k+1}=...=q_r=0$. Let

$$\hat{N}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \sqrt[-q_\kappa]{\phi_{\textnormal{soft}}(t)}.$$

Then $\hat{N}$ is regular and $\kappa$-soft. For $(t,z) \in H$ with $\hat{N}(t)<\vert{z_\kappa(t,z)}\vert<1$ we see that the inequality in $(+)$ is satisfied (since $\phi(t) \neq 0$). By Proposition 5.67(2) there is a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $M^*:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M^*) \subset \mathcal{A}(z_\kappa,\hat{N},1)$. So take $M:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{M^*(t),M^+(t)\}$. With (+) we see that $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$.

Case 2:

Assume $\kappa=k$ and $(q_{k+1},...,q_r) \neq 0$. Let $s:=-(2-(1/2)^{r-k})q_k$. Set

$$\hat{N}:\pi(C) \to \mathbb{R}_{>0}, t \mapsto \sqrt[s]{\phi_{\textnormal{soft}}(t)},$$

$$\hat{M}:\pi(C) \to \mathbb{R}_{>0}, t \mapsto 1/\exp_{r-k}(m^2 \pi).$$

Then $\hat{N}$ is $k$-soft and regular.

Let $(t,z) \in \mathcal{A}(z_\kappa,\hat{N},\hat{M})$. Then $\phi(t) \neq 0$. We show that the inequality in $(+)$ is satisfied for this $(t,z)$. We obtain

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} \vert{z_k(t,z)}\vert^{\tfrac{1}{2^j}} > \sqrt[q_k]{\phi(t)}$$

and therefore

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} (1/\vert{z_k(t,z)}\vert)^{-\tfrac{1}{2^j}} > \sqrt[q_k]{\phi(t)}.$$

We obtain

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} (1/\vert{z_k(t,z)}\vert)^{-\bigl|\tfrac{q_{k+j}}{2^j m q_k}\bigl|} > \sqrt[q_k]{\phi(t)}$$

since $1/\vert{z_k(t,z)}\vert>\exp_{r-k}(m^2 \pi)$ and $\vert{\tfrac{q_{k+j}}{m q_k}}\vert \leq 1$. With Claim 1 applied to $1/\sigma_kz_k(t,z)$ we obtain

$$\vert{\log_j(1/\sigma_kz_k(t,z))}\vert < \sqrt[2^jm]{1/\vert{z_k(t,z)}\vert}$$

for $j \in \{1,...,r-k\}$ and therefore

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} \vert{\log_j(1/\sigma_kz_k(t,z))}\vert^{-\vert{q_{k+j}/q_k}\vert} > \sqrt[q_k]{\phi(t)}.$$

Because $\vert{\log_j(1/\sigma_kz_k(t,z))}\vert>1$ for every $j \in \{1,...,r-k\}$ we obtain

$$\vert{z_k(t,z)}\vert \prod_{j=1}^{r-k} \vert{\log_j(1/\sigma_kz_k(t,z))}\vert^{q_{k+j}/q_k} > \sqrt[q_k]{\phi(t)}.$$

Now define $N$ and $M$ analoguously as in Case 1 (outgoing from $\hat{N}$ and $\hat{M}$ instead from $\hat{N}$ and $1$). With $(+)$ we have $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$.

All in all we are done with the proof of Proposition 5.81.

5.82 Corollary

Let $T:\pi(C) \to \mathbb{R}$ be $\kappa$-persistent. Suppose that $T_{\textnormal{soft}}$ is $\kappa$-soft. Let $q \in \mathbb{Q}^{r+1}$ be $\kappa$-negative. Then there is a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ such that

$$\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T).$$

Note that $\sqrt{T}$ is $\kappa$-persistent and that $\sqrt{T_{\textnormal{soft}}}$ is $\kappa$-soft. Let $q_\kappa^*:=\vert{q_\kappa}\vert+1$. Let

$$\hat{\mathcal{Z}}:=(\sigma_\kappa z_\kappa)^{q_\kappa-q_\kappa^*}\prod_{j \neq \kappa}(\sigma_jz_j)^{q_j}.$$

Then we have $\mathcal{B}((\sigma_\kappa z_\kappa)^{q_\kappa^*},\sqrt{T}) \cap \mathcal{B}(\hat{\mathcal{Z}},\sqrt{T}) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$. With Proposition 5.81(1) respectively Proposition 5.81(3) we find regular $\kappa$-persistent functions $M_1,M_2:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft function $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that
$\mathcal{B}(\mathcal{P}_{0,\kappa},M_1) \subset \mathcal{B}((\sigma_\kappa z_\kappa)^{q^*_\kappa},\sqrt{T})$ and $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M_2) \subset \mathcal{B}(\hat{\mathcal{Z}},\sqrt{T})$. Set $M:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{M_1(t),M_2(t)\}.$ We see that $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes q},T)$.

5.83 Convention

Let $D \subset \pi(C)$ be definable. If a function $g:D \to \mathbb{R}$ is the restriction of a regular (consistent) function then we call $g$ also regular (consistent).

For the next proposition we set the following: For a definable function $W:Y \to \mathbb{R}_{\geq 0}$ where $Y \subset \pi(C)$ is definable we set

$$W^C: \pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \left\{\begin{array}{ll} W(t), & t \in Y, \\
0, & \textnormal{ else.}\end{array}\right.$$

For $0 < l \leq m$ we denote by $M_l(s \times (r+1),\mathbb{Q})$ the set of all $(s \times (r+1))$-matrices whose first $l$ columns are zero.

Now we are ready to establish the desired preparation result.

5.84 Proposition

Let $g \in \log(E) \cup \{f\}$. There is a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},N,M) \subset \Lambda_g$ and $G:=\Phi_g|_{\mathcal{A}(\mathcal{P}_{0,\kappa},N,M)}$ is complex $r$-log-analytically prepared in $z$ with a complex LA-preparing tuple

where $q_G:=((q_G)_0,...,(q_G)_r)$, $b_G:=(b_{1,G},...,b_{s_G,G})$ and the following holds.

(1) We have $(q_G)_0,...,(q_G)_{\kappa-1}=0$ and $P_G \in M_\kappa(s \times (r+1), \mathbb{Q})$.

(2) $a_G$ is regular. If $g \in \log(E)$ then the following holds: $a_G$ is consistent and if $q_G$ is $\kappa$-negative then $1/((\vert{a_G}\vert^C)_{\textnormal{soft}})$ is $\kappa$-soft.

(3) $b_G$ is regular, consistent and if $p_{i,G}$ is $\kappa$-negative for $i \in \{1,...,s_G\}$ then $1/((\vert{b_{i,G}}\vert^C)_{\textnormal{soft}})$ is $\kappa$-soft where $P_G:=(p_{1,G},...,p_{s_G,G})^t$.

Let $g$ be log-exp-analytically $(l,r)$-prepared in $x$ with respect to $E$ for $l \in \{-1,...,e\}$. We do an induction on $l$. For $l=-1$ there is nothing to show.

$l-1 \to l:$ Let

where $b_g:=(b_1,...,b_s)$, $\exp(d_g):=(\exp(d_1),...,\exp(d_s))$, $P_g:=(p_1,...,p_s)^t$ and $\vert{v_g}\vert<\pi$. We may assume that $a \neq 0$. Otherwise the statement is clear. Note that $d_i$ is locally bounded in $(u,x)$ with reference set $X$ for every $i \in \{0,...,s\}$. For $i \in \{0,...,s\}$ let $\Lambda_i:=\Lambda_{d_i}$. Let

be a complex preparing tuple for $\Phi_g$ where $\tilde{D}_j:=\Phi_{d_j}$ for $j \in \{0,...,s\}$ and $\exp(\tilde{D}):=(\exp(\tilde{D}_1),...,\exp(\tilde{D}_s))$.

By the inductive hypothesis there is a regular $\kappa$-soft $N^*:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $M^*:\pi(C) \to \mathbb{R}_{\geq 0}$ such that for $i \in \{0,...,s\}$

$$B^*:= \mathcal{A}(\mathcal{P}_{0,\kappa},N^*,M^*) \subset \Lambda_i$$

and $D_i:=\tilde{D}_i|_{B^*}$ is complex $r$-log-analytically prepared in $z$ with complex LA-preparing tuple

where $q_{D_i}:=((q_{D_i})_0,...,(q_{D_i})_r)$, $P_{D_i}:=(p_{1,D_i},...,p_{s_{D_i},D_i})^t$, $b_{D_i}:=(b_{1,D_i},...,b_{s_{D_i},D_i})$ and the following properties (1) - (3) holds for this preparing tuple:

(1) $(q_{D_i})_0=...=(q_{D_i})_{\kappa-1}=0$ and $P_{D_i} \in M_{\kappa}(s_{D_i} \times (r+1), \mathbb{Q})$.

(2) $a_{D_i}$ is regular. If $g \in \log(E)$ the following holds: $a_{D_i}$ is consistent and if $q_{D_i}$ is $\kappa$-negative then $1/((\vert{a_{D_i}}\vert^C)_{\textnormal{soft}})$ is $\kappa$-soft.

(3) $b_{D_i}$ is regular, consistent and if $p_{j,D_i}$ is $\kappa$-negative then $1/((\vert{b_{j,D_i}}\vert^C)_{\textnormal{soft}})$ is $\kappa$-soft for $j \in \{1,...,s_{D_i}\}$.

At first we deal with $\exp(D_0),...,\exp(D_s)$ and start with the following

Claim 1

Let $0<\delta<1/2$. Then there is a regular $\kappa$-soft $S_\delta:\pi(C) \to \mathbb{R}_{\geq 0}$  and a regular $\kappa$-persistent $T_\delta:\pi(C) \to \mathbb{R}_{\geq 0}$ with $Y_\delta:=\mathcal{A}(\mathcal{P}_{0,\kappa},S_\delta,T_\delta) \subset B^*$ and for $i \in \{0,...,s\}$ definable functions $D_{i,1}:\pi(Y_\delta) \to \mathbb{R}, t \mapsto D_{i,1}(t),$ and $D_{i,2}:Y_\delta \to \mathbb{C}, (t,z) \mapsto D_{i,2}(t,z),$ with the following properties.

(2)* $D_{i,1}$ is regular and consistent.

(3)* $D_{i,2}$ is bounded by $\delta$. Additionally $\exp(D_{i,2})$ is complex $r$-log-analytically prepared in $z$ with complex LA-preparing tuple $(r,\mathcal{Z},a^*,q^*,s^*,V^*,b^*,P^*)$ with which fulfills properties (1) - (3) above where $a^*=1$ and $q^*=0$.

Proof of Claim 1

Let $K \in \{D_0,...,D_s\}$. It suffices to find a regular $\kappa$-soft $\hat{S}_\delta:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $\hat{T}_\delta:\pi(C) \to \mathbb{R}_{\geq 0}$  with $W:= \mathcal{A}(\mathcal{P}_{0,\kappa},\hat{S}_{\delta},\hat{T}_{\delta}) \subset B^*$ and corresponding functions $K_1:\pi(W) \to \mathbb{R}$ and $K_2:W \to \mathbb{C}$ which fulfill properties (1)* - (3)*. Let

Suppose $\mathfrak{a} \neq 0$. Otherwise the claim follows. Fix $R>1$ such that $\mathfrak{V}$ converges absolutely on $\overline{D^\mathfrak{s}(0,R)}$. Let $\sum_{\alpha \in \mathbb{N}_0^{\mathfrak{s}}} c_\alpha Z^{\alpha}$ be the power series expansion of $\mathfrak{V}$ where $Z:=(Z_1,...,Z_s)$. Fix $L \in \mathbb{R}_{>0}$ with $\delta<L$ such that $\sum_{\alpha \in \mathbb{N}^{\mathfrak{s}}}\vert{c_\alpha w^{\alpha}}\vert<L$ for $w \in D^\mathfrak{s}(0,R)$. We are doing two cases.

1. Case: $(\mathfrak{q}_\kappa,...,\mathfrak{q}_r) \neq 0$.

Consider the regular function

$$T_\delta^*: \pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \left\{\begin{array}{ll} \min\{\frac{\delta}{L \vert{\mathfrak{a}(t)}\vert},\frac{1}{\vert{\mathfrak{a}(t)}\vert}\} , & t \in \pi(B^*),\\
0, & \textnormal{else}. \end{array}\right.$$

Subclaim 1

$T_\delta^*$ is $\kappa$-persistent.

Proof of Subclaim 1

Let $w \in \mathbb{R}^l$. Let $\gamma:\textnormal{}]0,1[\textnormal{} \to C_w$ be a definable curve compatible with $C_w$ with $\lim_{y \searrow 0} \gamma(y) \in X_w$ and $\lim_{y \searrow 0} \vert{\gamma_x(y)-\mu_\kappa(w,\gamma_u(y))}\vert=0$. Note that there is $0<\epsilon \leq 1$ such that $(w,\gamma_u(y)) \in \pi(B^*)$ for every $y \in \textnormal{}]0,\epsilon[$ (since $N^*$ is $\kappa$-soft and $M^*$ is $\kappa$-persistent). Since $\mathfrak{a}$ is consistent we see that $\lim_{y \searrow 0}T_\delta^*(w,\gamma_u(y)) \in \mathbb{R}_{>0}$. $\blacksquare_{\textnormal{Subclaim }1}$

Note that if $\vert{(\sigma\mathcal{Z})^{\otimes \mathfrak{q}}(t,z)}\vert < T_\delta^*(t)$ then $\vert{K(t,z)}\vert < \delta$ for $(t,z) \in B^*$.

1.1 Case: $\mathfrak{q}$ is $\kappa$-positive.

By Proposition 5.81(1) if $q_\kappa > 0$ respectively Proposition 5.81(2) if $\kappa=k \geq 0$ and $q_k=0$ (since $(T_\delta^*)_{\textnormal{pers}}$ is consistent, because $\mathfrak{a}$ is consistent) there is a regular $\kappa$-persistent $M_\delta^+:\pi(C) \to \mathbb{R}_{ \geq 0}$ such that $\mathcal{B}(\mathcal{P}_{0,\kappa},M_\delta^+) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes \mathfrak{q}},T_\delta^*)$. Take $\hat{S}_\delta:=N^*$,

$$\hat{T}_\delta:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{M_\delta^+(t),M^*(t)\},$$

$K_1:=0$ and $K_2:=K|_W$ where $W:=\mathcal{A}(\mathcal{P}_{0,\kappa},\hat{S}_\delta,\hat{T}_\delta)$. Note that $\exp(K_2)=\exp^*(K_2)$ where

$$\exp^*:\mathbb{C} \to \mathbb{C}, z \mapsto \left\{\begin{array}{ll} \exp(z), & \vert{z}\vert \leq \delta,\\
0, & \textnormal{ else,}\end{array}\right.$$

is globally subanalytic. Additionally we obtain $\vert{\mathfrak{a}(t)(\sigma\mathcal{Z})^{\otimes \mathfrak{q}}(t,z)}\vert < 1$ for $(t,z) \in W$. By composition of power series we get the desired preparation for $\exp(K_2)$.

1.2 Case: $\mathfrak{q}$ is $\kappa$-negative.

Similarly as in the proof of Subclaim 1 we see that $(T_\delta^*)_{\textnormal{soft}}$ is $\kappa$-soft (since $1/((\vert{\mathfrak{a}}\vert^C)_{\textnormal{soft}})$ is $\kappa$-soft). By Corollary 5.82 there is a regular $\kappa$-soft $N_\delta^+:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $M_\delta^+:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},N_\delta^+,M_\delta^+) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes \mathfrak{q}},T_\delta^*)$. Take $\hat{S}_\delta:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \max\{N_\delta^+(t),N^*(t)\}$, $\hat{T}_\delta:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{M_\delta^+(t),M^*(t)\}$, $K_1:=0$ and $K_2:=K|_W$ where $W=\mathcal{A}(\mathcal{P}_{0,\kappa},\hat{S}_\delta,\hat{T}_\delta)$. We are done in a completely similar way as in Case 1.1.

2. Case: $(\mathfrak{q}_\kappa,...,\mathfrak{q}_r) = 0$.

$$\mathcal{S}:=\{i \in \{1,...,\mathfrak{s}\} \mid (\mathfrak{p}_{i\kappa},...,\mathfrak{p}_{ir}) \neq 0\}.$$

If $\mathcal{S}=\emptyset$ then $K=\mathfrak{a} \mathfrak{V}(\mathfrak{b}_1,...,\mathfrak{b}_\mathfrak{s})$ is a regular and consistent function considered as function on $\pi(B^*)$. So we are done by taking $\hat{S}_\delta:=N^*$, $\hat{T}_\delta:=M^*$, $K_1:=0$ and $K_2:=K$. So suppose $\mathcal{S} \neq \emptyset$. Let

$$\Gamma_1:=\{\alpha \in \mathbb{N}_0^{\mathfrak{s}} \mid {^t}P \alpha = 0\}$$

and $\Gamma_2:= \mathbb{N}_{0}^\mathfrak{s} \setminus \Gamma_1.$ Note that $\Gamma_2 \neq \emptyset$. Set $\mathfrak{V}_1:=\sum_{\alpha \in \Gamma_1}c_{\alpha}Z^{\alpha}$ and $\mathfrak{V}_2:=\sum_{\alpha \in \Gamma_2}c_{\alpha}Z^{\alpha}$. For $l \in \{1,2\}$ let

$$J_l :=  \mathfrak{a} \mathfrak{V}_l(\mathfrak{b}_1(\sigma\mathcal{Z})^{\otimes \mathfrak{p}_1},...,\mathfrak{b}_\mathfrak{s}(\sigma\mathcal{Z})^{\otimes \mathfrak{p}_\mathfrak{s}}).$$

Note that $K=J_1+J_2$. Let $\phi_i:=\mathfrak{b}_i(\sigma\mathcal{Z})^{\otimes \mathfrak{p}_i}$ for $i \in \{1,...,\mathfrak{s}\}$. For $j \in \mathcal{S}$ we define the regular function

$$U_{j,\delta}:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \left\{\begin{array}{ll} \bigl|\frac{\delta}{L \mathfrak{a}(t)\mathfrak{b}_j(t)} \bigl|, & t \in \pi(B^*),\\
0, & \textnormal{else}. \end{array}\right. $$

Similarly as in the proof of Subclaim 1 one sees for $j \in \mathcal{S}$ that $U_{j,\delta}$ is $\kappa$-persistent (since $\mathfrak{b}_j$ and $\mathfrak{a}$ are consistent) and that if $\mathfrak{p}_j$ is $\kappa$-negative then $(U_{j,\delta})_{\textnormal{soft}}$ is $\kappa$-soft (since $\mathfrak{a}$ is consistent and $1/((\vert{\mathfrak{b}_j}\vert^C)_{\textnormal{soft}})$ is $\kappa$-soft). Additionally $(U_{j,\delta})_{\textnormal{pers}}$ is consistent for $j \in \mathcal{S}$.

We see that for $j \in \mathcal{S}$ there is a regular $\kappa$-soft $\hat{S}_{j,\delta}:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $\hat{T}_{j,\delta}:\pi(C) \to \mathbb{R}_{\geq 0}$ such that

$$\mathcal{A}(\mathcal{P}_{0,\kappa},\hat{S}_{j,\delta},\hat{T}_{j,\delta}) \subset \mathcal{B}((\sigma\mathcal{Z})^{\mathfrak{p}_j},U_{j,\delta}).$$

(By Proposition 5.81(1) if $\mathfrak{p}_j$ is $\kappa$-positive and $\mathfrak{p}_{j \kappa} \neq 0$ respectively Proposition 5.81(2) if $\mathfrak{p}_j$ is $\kappa$-positive, $\kappa=k \geq 0$ and $\mathfrak{p}_{jk} = 0$ since $(U_{j,\delta})_{\textnormal{pers}}$ is consistent respectively Corollary 5.82 if $\mathfrak{p}_j$ is $\kappa$-negative. If $\mathfrak{p}_j$ is $\kappa$-positive take $\hat{S}_{j,\delta}=0$.)
Of course we may assume that $N^* \leq \hat{S}_{j,\delta}$ and $\hat{T}_{j,\delta} \leq M^*$ for $j \in \mathcal{S}$. Consider $\hat{S}_\delta :\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \max\{ \hat{S}_{j,\delta}(t) \mid j \in \mathcal{S}\}$ and $\hat{T}_\delta :\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{ \hat{T}_{j,\delta}(t) \mid j \in \mathcal{S}\}$. Then $\hat{S}_\delta$ is regular and $\kappa$-soft and $\hat{T}_\delta$ is regular and $\kappa$-persistent. Consider $W:=\mathcal{A}(\mathcal{P}_{0,\kappa},\hat{S}_\delta,\hat{T}_\delta)$.

To obtain Claim 1 it remains to show that $\exp(J_2)|_W$ is complex $r$-log-analytically prepared in $z$ as desired (see Subclaim 2) and that $J_1|_W$ coincides with a regular and consistent function on $\pi(W)$.

Subclaim 2

For $(t,z) \in W$ we have $\vert{J_2(t,z)}\vert<\delta$. The function $\exp(J_2)|_W$ is complex $r$-log-analytically prepared in $z$ as required in property (3)*.

Proof of Subclaim 2

Set $\phi:=(\phi_1,...,\phi_\mathfrak{s})$. For every $\alpha:=(\alpha_1,...,\alpha_\mathfrak{s}) \in \Gamma_2$ fix $i_\alpha \in \{1,...,\mathfrak{s}\}$ with $i_\alpha \in \mathcal{S}$ and $\alpha_{i_{\alpha}} \neq 0$. Therefore we have for $(t,z) \in W$

We obtain

\begin{eqnarray*}
\vert{J_2(t,z)}\vert &=& \vert{\mathfrak{a}(t)\mathfrak{V}_2(\phi(t,z))}\vert\\
&=& \vert{\mathfrak{a}(t) \sum_{\alpha \in \Gamma_1}c_{\alpha} \phi_{i_{\alpha}}(t,z)^{\alpha_{i_\alpha}} \prod_{j \neq i_{\alpha}} \phi_j(t,z)^{\alpha_j}}\vert\\
&=&\vert{\sum_{\alpha \in \Gamma_1}c_{\alpha}\mathfrak{a}(t)\phi_{i_{\alpha}}(t,z) \phi_{i_{\alpha}}(t,z)^{\alpha_{i_\alpha}-1} \prod_{j \neq i_{\alpha}} \phi_j(t,z)^{\alpha_j}}\vert\\
&<& \delta/L \sum_{\alpha \in \Gamma_1}\vert{c_{\alpha} \phi_{i_{\alpha}}(t,z)^{\alpha_{i_\alpha}-1} \prod_{j \neq i_{\alpha}} \phi_j(t,z)^{\alpha_j}}\vert\\
&<& \delta.
\end{eqnarray*}

Let $\mathcal{S}:=\{j_1,...,j_\beta\}$ where $\beta \in \mathbb{N}$. Note that $i_\alpha \in \{j_1,...,j_\beta\}$ is unique for $\alpha \in \mathbb{N}_0^\mathfrak{s}$. Let $(w_{j_1},...,w_{j_\beta})$ be a new tuple of complex variables. Consider for $\gamma \in \{1,...,\beta\}$

$$\tilde{\phi}_\gamma:W \to \overline{B(0,1)}, (t,z) \mapsto a(t)b_{j_\gamma}(t)(\sigma\mathcal{Z})^{\otimes \mathfrak{p}_{j_\gamma}}(t,z).$$

$$\hat{V}:\overline{D^{\mathfrak{s}+\beta}(0,1)} \to \mathbb{C}, (z_1,...,z_s,w_{j_1},...,w_{j_\beta}) \mapsto \sum_{\alpha \in \Gamma_2} c_{\alpha}w_{i_\alpha}z_{i_\alpha}^{\alpha_{i_\alpha}-1} \prod_{j \neq i_\alpha} z_j^{\alpha_j}.$$

Note that $\hat{V}$ defines a power series which converges absolutely on an open neighbourhood of $\overline{D^{\mathfrak{s}+\beta}(0,1)}$ (and is therefore globally subanalytic). We have for $(t,z) \in W$

and therefore

$$\exp^*:\mathbb{C} \to \mathbb{C}, z \mapsto \left\{\begin{array}{ll} \exp(z), & \vert{z}\vert \leq \delta, \\
0, & \textnormal{else}. \end{array}\right.$$

By using the exponential series and composition of power series we see that $\exp(J_2)$ has the desired properties. $\blacksquare_{\textnormal{Subclaim }2}$

$$\Xi:\pi(W) \to \mathbb{R}, t \mapsto \sum_{\alpha \in \Gamma_1}\mathfrak{a}(t) c_\alpha \prod_{j=1}^{\mathfrak{s}}\mathfrak{b}_j(t)^{\alpha_j}.$$

Then we obtain
$J_1(t,z)=\Xi(t)$ for $(t,z) \in W$. We show that $\Xi$ is consistent and regular and are done with Claim 1. Note that

$$\Xi(t)=\mathfrak{a}(t)\mathfrak{V}_1(\mathfrak{b}_1(t)(\sigma\mathcal{Z})^{\otimes \mathfrak{p}_1}(t,\lambda(t)),...,\mathfrak{b}_\mathfrak{s}(t)(\sigma\mathcal{Z})^{\otimes \mathfrak{p}_\mathfrak{s}}(t,\lambda(t)))$$

for $t \in \pi(W)$ where

$$\lambda:\pi(W) \to \mathbb{R}, t \mapsto \mu_\kappa(t)+(\prod_{j=0}^\kappa \sigma_j)\frac{\hat{S}_\delta(t)+\hat{T}_\delta(t)}{2},$$

is regular. (Note also that $(t,\lambda(t)) \in W$ for $t \in \pi(W)$.) So we see that $\Xi$ is consistent since $\mathfrak{a}$ is consistent.

Subclaim 3

The function $\Xi$ is regular.

Proof of Subclaim 3

Similarly as in the proof of the claim in Remark 5.79 we find a globally subanalytic function $G:\mathbb{C}^{s+1} \times \mathbb{C}^{r-\kappa+1} \to \mathbb{C}$ such that for $t \in \pi(W)$

where $\mathcal{Z}_\kappa:=(z_\kappa,...,z_r)$ and $\eta:\pi(W) \to \mathbb{R}^{s+1}, t \mapsto (\mathfrak{a}(t),\mathfrak{b}_1(t),...,\mathfrak{b}_s(t))$. Note that $\eta$ and $\Theta$ are regular. By Remark 5.55(2) it is enough to show that $\pi(C) \to \mathbb{C}, t \mapsto z_l(t,\lambda(t)),$ is regular. We do an induction on $l \in \{0,...,r\}$.

$l=0$: We have $z_0(t,\lambda(t))=\lambda(t)-\Theta_0(t)$ for $t \in \pi(W)$. Remark 5.55(2) gives the result since $\Theta_0$ is regular.

$l-1 \to l$:  We have $z_l(t,\lambda(t))=\log(\sigma_{l-1}z_{l-1}(t,\lambda(t)))-\Theta_l(t)$ for $t \in \pi(W)$. Remark 5.55(2) and the inductive hypothesis gives the result since $\Theta_l$ is regular.

$\blacksquare_{\textnormal{Subclaim }3}$

So we set $K_1:=\Xi$ and $K_2:=J_2|_W$.

All in all we obtain Claim 1.
$\blacksquare_{\textnormal{Claim }1}$

Let $Y_{\delta}$ and $D_{i,j}$ as in Claim 1 for $0<\delta<\pi$. We obtain for $(t,z) \in \Lambda_g \cap Y_\delta$

$$\Phi_g(t,z)=a(t)\exp(D_{0,1}(t))(\sigma\mathcal{Z})^{\otimes q}(t,z)\exp(D_{0,2}(t,z))$$

$$V(b_1(t)\exp(D_{1,1}(t))(\sigma\mathcal{Z})^{\otimes p_1}(t,z)\exp(D_{1,2}(t,z)),...,$$

$$b_s(t)\exp(D_{s,1}(t))(\sigma\mathcal{Z})^{\otimes p_s}(t,z)\exp(D_{s,2}(t,z)))$$

Note that $q_l \neq 0$ for $l \in \{0,...,\kappa-1\}$ and $p_{ij} \neq 0$ for $i \in \{1,...,s\}$ and $j \in \{0,...,\kappa-1\}$ in general. So we deal with $\prod_{j=0}^{\kappa-1}(\sigma_jz_j)^{q_j}$ respectively with $\prod_{j=0}^{\kappa-1}(\sigma_jz_j)^{p_{ij}}$ in the following

Claim 2

Let $1<\epsilon$ and $p_0:=(p_{00},...,p_{0r}):=q$. There is a regular $\kappa$-persistent $T_\epsilon:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},N^*,T_\epsilon) \subset B^*$ and the following holds for $i \in \{0,...,s\}$.

The function $\omega_i:=\prod_{j=0}^{\kappa-1} (\sigma_jz_j)^{p_{ij}}|_{\mathcal{A}(\mathcal{P}_{0,\kappa},N^*,T_\epsilon)}$ is complex $r$-log-analytically prepared in $z$ with complex LA-preparing tuple

such that $a_{w_i} \neq 0$, $a_{w_i}$ and $b_{w_i}$ are regular and consistent, $q_{\omega_i} = 0$, $(P_{\omega_i})_{l(\kappa+1)} = 1$ and $(P_{\omega_i})_{lj}=0$ for $j \in \{1,...,r+1\} \setminus \{\kappa+1\}$ and $l \in \{1,...,s_{\omega_i}\}$, and $\vert{V_{w_i}(\overline{D^{s_{\omega_i}}(0,1)})}\vert \subset [1/\epsilon,\epsilon]$.

Let $\nu \in \{p_0,...,p_s\}$. It suffices to find a regular $\kappa$-persistent $\hat{T}_\epsilon:\pi(C) \to \mathbb{R}_{\geq 0}$ with $\mathcal{A}(\mathcal{P}_{0,\kappa},N^*,\hat{T}_\epsilon) \subset B^*$ such that $\prod_{j=0}^{\kappa-1}(\sigma_jz_j)^{\nu_j}|_{\mathcal{A}(\mathcal{P}_{0,\kappa},N^*,\hat{T}_\epsilon)}$ fulfills the desired properties.

Let $0 < \beta \leq 1/2$ be such that $1/\epsilon<\prod_{j=0}^{\kappa-1}\vert{1+z}\vert^{\nu_j}<\epsilon$ for $z \in B(0,\beta)$. By Proposition 5.67(1) there is a regular $\kappa$-persistent $T^*_\beta:\pi(C) \to \mathbb{R}_{\geq 0}$ such that

$$\Bigl |\frac{\mathcal{P}_{j,\kappa}(t,z)}{e^{\mu_{\kappa-j-1,\kappa}(t)}} \Bigl| < \beta$$

for every $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},T^*_\beta)$ and every $j \in \{0,...,\kappa-1\}$. We have

$$\prod_{j=0}^{\kappa-1}(\sigma_jz_j(t,z))^{\nu_j}=\prod_{j=0}^{\kappa-1} e^{\nu_j\mu_{\kappa-j-1,\kappa}(t)} \bigl(1+\frac{\sigma_j\mathcal{P}_{j,\kappa}(t,z)}{e^{\mu_{\kappa-j-1,\kappa}(t)}} \bigl)^{\nu_j} \textnormal{ }(+)$$

for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},T^*_\beta)$ (compare with Proposition 5.18).

We say that a function $S:Z \to \mathbb{C}$ where $Z \subset \mathcal{B}(\mathcal{P}_{0,\kappa},T^*_\beta)$ is definable fulfills property $(*)_{\textnormal{prep}}$ if the following holds:

$J$ is complex $r$-log-analytically prepared in $z$ with complex LA-preparing tuple

such that $a_J \neq 0$, $a_J$ and $b_J$ are regular and consistent, $(q_J)_{\kappa+1}=1$ and $(q_J)_j=0$ for $j \in \{1,...,r+1\} \setminus \{\kappa+1\}$, $(P_J)_{i(\kappa+1)}=1$ and $(P_J)_{ij}=0$ for $j \in \{1,...,r+1\} \setminus \{\kappa+1\}$ and $i \in \{1,...,s_J\}$.

$$\omega: \mathbb{R}_{>0} \to ]0,1/2], r \mapsto \sup\{c \in \textnormal{} ]0,1/2[ \textnormal{}\mid \vert{\exp^*(z)-1}\vert < r \textnormal{ for every }z \in B(0,c)\}$$

$$\exp^*:\mathbb{C} \to \mathbb{C}, z \mapsto \left\{\begin{array}{ll} \exp(z), & \vert{z}\vert \in [-1/2,1/2], \\
0, & \textnormal{else}. \end{array}\right.$$

Since $\exp^*$ is globally subanalytic we see that $\omega$ is globally subanalytic. Note that if $\vert{z}\vert<\omega(a)$ then $\vert{\exp^*(z)-1}\vert<a$ for $z \in \mathbb{C}^-$ and $a \in \mathbb{R}_{>0}$. Let $\Omega_\kappa = M_\delta$ and by a descending induction on $l \in \{1,...,\kappa\}$ let $\Omega_{l-1}:\pi(C) \to \textnormal{}]0,1/2], t \mapsto \omega(\Omega_l(t)/e^{\mu_{l-1,\kappa}(t)})$. Note that $\Omega_l$ is regular and $\kappa$-persistent for $l \in \{0,...,\kappa\}$. Let $\Omega:=\Omega_0$ and $D:=\mathcal{B}(z_\kappa,\Omega)$.

Let $E_0:=z_\kappa|_D$. Inductively for $l \in \{1,...,\kappa\}$ let

$$E_l:D \to \mathbb{C}, (t,z) \mapsto \sigma_{k-l}e^{\mu_{l-1,\kappa}(t)}(e^{E_{l-1}(t,z)}-1).$$

We see by an easy induction on $l \in \{0,...,\kappa\}$ that $E_l(t,z) = \mathcal{P}_{\kappa-l,\kappa}(t,z)$ for $(t,z) \in D$. For $(t,z) \in D$ we obtain

\begin{eqnarray*}
\vert{E_{l-1}(t,z)}\vert<\Omega_{l-1}(t) &\Rightarrow& \vert{e^{E_{l-1}(t,z)} - 1}\vert < \Omega_l(t)/e^{\mu_{l-1,\kappa}(t)} \\
&\Rightarrow& \vert{\sigma_{k-l}e^{\mu_{l-1,\kappa}(t)}(e^{E_{l-1}(t,z)} - 1)}\vert<\Omega_l(t),
\end{eqnarray*}

i.e. $\vert{E_l(t,z)}\vert<\Omega_l(t)$. This shows that

$$\vert{\mathcal{P}_{0,\kappa}(t,z)}\vert = \vert{E_\kappa(t,z)}\vert < \Omega_\kappa(t) = M_\delta$$

for $(t,z) \in D$ and therefore $D \subset \mathcal{B}(\mathcal{P}_{0,\kappa},M_\delta)$.
Let $\Omega:=T_\beta^*$.

Subclaim 1

By shrinking $\Omega$ if necessary we have that $\mathcal{P}_{0,\kappa}|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)}$ fulfills property $(*)_{\textnormal{prep}}$ (i.e. there is a regular $\kappa$-persistent $\hat{\Omega}:\pi(C) \to \mathbb{R}_{\geq 0}$ with $\hat{\Omega} \leq \Omega$ such that $\mathcal{P}_{0,\kappa}|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\hat{\Omega})}$ fulfills property $(*)_{\textnormal{prep}}$).

Proof of Subclaim 1

Let $E_0:=z_\kappa|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)}$. Inductively for $l \in \{1,...,\kappa\}$ let

$$E_l:\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega) \to \mathbb{C}, (t,z) \mapsto \sigma_{k-l}e^{\mu_{l-1,\kappa}(t)}(e^{E_{l-1}(t,z)}-1).$$

We see by an easy induction on $l \in \{0,...,\kappa\}$ that $E_l(t,z) = \mathcal{P}_{\kappa-l,\kappa}(t,z)$ for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)$. We show by induction on $l \in \{0,...,\kappa\}$ that $E_l|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)}$ fulfills property $(*)_{\textnormal{prep}}$ by shrinking $\Omega$ if necessary. For $l=0$ this is clear.

$l-1 \to l:$ Let

be a corresponding complex LA-preparing tuple for $E_{l-1}|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)}$ where $\tilde{P}:=(\tilde{p}_{i(j-1)})_{i,j}$. We have for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)$

$$E_l(t,z)=\sigma_{k-l}e^{\mu_{l-1,\kappa}(t)}(\exp(\tilde{a}(t)\sigma_\kappa z_\kappa(t,z)Q(t,z)) - 1)$$

$$Q(t,z) := \tilde{V}(\tilde{b}_1(t)\sigma_\kappa z_\kappa(t,z),...,\tilde{b}_{\tilde{s}}(t)\sigma_\kappa z_\kappa(t,z)).$$

We see

$$E_l(t,z)=\sigma_{k-l}e^{\mu_{l-1,\kappa}(t)}\tilde{a}(t)\sigma_\kappa z_\kappa(t,z) \frac{\exp(\tilde{a}(t)\sigma_\kappa z_\kappa(t,z)Q(t,z))-1}{\tilde{a}(t)\sigma_\kappa z_\kappa(t,z)}.$$

Note that $\tilde{a} \neq 0$. Because $\tilde{a}$ is consistent and regular we see that $1/\vert{\tilde{a}}\vert$ is regular and $\kappa$-persistent. So we may assume that $\vert{z_\kappa(t,z)}\vert<1/\vert{\tilde{a}(t)}\vert$ for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)$ after shrinking $\Omega$ if necessary (by Proposition 5.81(1) there is a regular $\kappa$-persistent $\hat{M}:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{B}(\mathcal{P}_{0,\kappa},\hat{M}) \subset \mathcal{B}(z_\kappa,1/\vert{\tilde{a}}\vert)$). Note that $\sigma_{k-l}e^{\mu_{l-1,\kappa}}\tilde{a}$ is consistent and regular (since $e^{\mu_{l-1,\kappa}}$ is consistent and regular). Now it is easy to check by using the exponential series and composition of power series that $E_l$ is complex $r$-log-analytically prepared with a preparing tuple which fulfills the desired properties. $\blacksquare_{\textnormal{Subclaim }1}$

Subclaim 2

By shrinking $\Omega$ if necessary we have that $\mathcal{P}_{l,\kappa}|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)}$ fulfills property $(*)_{\textnormal{prep}}$ for $l \in \{0,...,\kappa-1\}$.

Proof of Subclaim 2

We do an induction on $l$. By Claim 1 we get the assertion for $l=0$.

$l-1 \to l:$ Let

be a corresponding complex LA-preparing tuple for $\mathcal{P}_{l-1,\kappa}|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)}$. Let $\alpha:=\hat{a}/e^{\mu_{\kappa-l,\kappa}}$. We have for $(t,z) \in D$

\begin{eqnarray*}
\mathcal{P}_{l,\kappa}(t,z)&=&\log^*(1+\frac{\mathcal{P}_{l-1,\kappa}(t,z)}{e^{\mu_{\kappa-l,\kappa}(t)}})\\
&=&\alpha(t)\sigma_\kappa z_\kappa(t,z) \frac{\log^*(1+\alpha(t)\sigma_\kappa z_\kappa(t,z)Q(t,z))}{\alpha(t)\sigma_\kappa z_\kappa(t,z)}
\end{eqnarray*}

where for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)$

$$Q(t,z):= \hat{V}(\hat{b}_1(t)\sigma_\kappa z_\kappa(t,z),...,\hat{b}_{\hat{s}}(t)\sigma_\kappa z_\kappa(t,z)).$$

Note that $\alpha$ is regular and consistent since $e^{\mu_{\kappa-l,\kappa}}$ is regular and consistent. So $1/\vert{\alpha}\vert$ is regular and $\kappa$-persistent. So by shrinking $\Omega$ if necessary we may assume that $\vert{z_\kappa(t,z)}\vert<1/\vert{\alpha(t)}\vert$ for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)$. We see by using the logarithmic series and composition of power series that $\mathcal{P}_{l,\kappa}|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)}$ fulfills property $(*)_{\textnormal{prep}}$. $\blacksquare_{\textnormal{Subclaim }2}$

So by Subclaim 2 we find for $j \in \{0,...,\kappa-1\}$ a complex preparing tuple

for $\mathcal{P}_{j,\kappa}|_{\mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)}$ according to property $(*)_{\textnormal{prep}}$ after shrinking $\Omega$ if necessary. Note that $a^*_j(t)/e^{\mu_{\kappa-j-1,\kappa}(t)}$ is regular and consistent. Therefore the absolute value of its reciprocal is regular and $\kappa$-persistent. So by further shrinking $\Omega$ if necessary we may assume that $\vert{z_\kappa(t,z)}\vert < e^{\mu_{\kappa-j-1,\kappa}(t)}/\vert{a_j^*(t)}\vert$ for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},\Omega)$ and $j \in\{0,...,\kappa-1\}$. So choose $T_\epsilon:=\Omega$. With (+) and composition of power series we obtain the desired preparation for $\prod_{j=0}^{\kappa-1}(\sigma_jz_j)^{\nu_j}|_{\mathcal{A}(\mathcal{P}_{0,\kappa},N^*,T_\epsilon)}$ since $\prod_{j=0}^{\kappa-1}e^{\nu_j \mu_{\kappa-j-1,\kappa}}$ is regular and consistent (by writing $\prod_{j=0}^{\kappa-1}(1+w_j)^{\nu_j}$ as a power series for $w:=(w_0,...,w_{\kappa-1}) \in D^\kappa(0,1/2)$ which converges absolutely on an open neighbourhood of $\overline{D^\kappa(0,1/2)}$). This gives Claim 2. $\blacksquare_{\textnormal{Claim }2}$

Fix $1<\epsilon < \sqrt[4]{R}$. To this $\epsilon$ fix the corresponding regular $\kappa$-persistent function $T_\epsilon$ from Claim 2. Then

$$\prod_{j=0}^{\kappa-1}(\sigma_jz_j)^{p_{ij}}|_{\mathcal{A}(\mathcal{P}_{0,\kappa},N^*,T_\epsilon)} = a_{w_i} V_{w_i}((b_{w_i})_1 \sigma_\kappa z_\kappa,...,(b_{w_i})_{s_{w_i}} \sigma_\kappa z_\kappa):=a_{w_i}Q_i$$

for $i \in \{0,...,s\}$ where $a_{w_i} \neq 0$, $a_{w_i}$ and $b_{w_i}$ are regular and consistent. Note that $1/\sqrt[4]{R} < \vert{Q_i(t,z)}\vert < \sqrt[4]{R}$ for $(t,z) \in \mathcal{B}(\mathcal{P}_{0,\kappa},T_\epsilon)$. (++)

Fix $0<\delta<\min\{\log(\sqrt[4]{R}),\pi\}$. To this $\delta$ fix the corresponding regular $\kappa$-soft $S_\delta$ and the regular $\kappa$-persistent $T_\delta$ from Claim 1. By shrinking $T_\delta$ if necessary we may assume that $T_\delta \leq T_\epsilon$.

Let $Y:=Y_\delta=\mathcal{A}(\mathcal{P}_{0,\kappa},S_\delta,T_\delta)$ (note that $Y \subset \mathcal{A}(\mathcal{P}_{0,\kappa},N^*,M^*)$) and let $D_{i,1}$ and $D_{i,2}$ be as in Claim 1 for $i \in \{0,...,s\}$. Then $1/\sqrt[4]{R}<\vert{\exp(D_{i,2}(t,z))}\vert<\sqrt[4]{R}$ for $(t,z) \in Y$ and $i \in \{0,...,s\}$. (+++)

For $\lambda:=(\lambda_0,...,\lambda_r) \in \mathbb{Q}^{r+1}$ and $(t,z) \in \Lambda_g \cap Y$ let

$(\sigma\mathcal{Z})_o^{\otimes \lambda}(t,z):=\prod_{j=\kappa}^r(\sigma_jz_j(t,z))^{\lambda_j}$. For $(t,z) \in \Lambda_g \cap Y$ we have

$$\Phi_g(t,z)=a(t)a_{\omega_0}(t)\exp(D_{0,1}(t))(\sigma\mathcal{Z})_o^{\otimes q}(t,z)Q_0(t,z)\exp(D_{0,2}(t,z))$$

$$V(b_1(t)a_{\omega_1}(t)\exp(D_{1,1}(t))(\sigma\mathcal{Z})_o^{\otimes p_1}(t,z)Q_1(t,z)\exp(D_{1,2}(t,z)),...,$$

$$b_s(t)a_{\omega_s}(t)\exp(D_{s,1}(t))(\sigma\mathcal{Z})_o^{\otimes p_s}(t,z)Q_s(t,z)\exp(D_{s,2}(t,z))).$$

Let $b_0:=a$ and for $i \in \{0,...,s\}$

$$A_i: \pi(\Lambda_g \cap Y) \to \mathbb{R}, t \mapsto b_i(t)a_{\omega_i}(t)\exp(D_{i,1}(t)).$$

Note that for $i \in \{1,...,s\}$ the function $A_i$ is consistent (by Remark 5.52(3) $\exp(D_{i,1})$ is consistent, by Remark 5.53(1) $b_i$ is also consistent) and regular (by Remark 5.55(2) since $b_i$ is $C$-nice and therefore regular). Note that $A_0$ is regular (since $a$ is $C$-nice and therefore regular), and if $g \in \log(E)$ then $A_0$ is consistent (by Remark 5.53(2) $b_0$ is consistent if $g \in \log(E)$, by Remark 5.52(3) $\exp(D_{0,1})$ is consistent).

Note that $\vert{\textnormal{Im}(D_{i,2}(t,z))}\vert<\delta < \pi$ for $(t,z) \in Y$ and $i \in \{0,...,s\}$. So we have

$$\Lambda_g \cap Y = \{(t,z) \in Y \mid \vert{A_i(t)(\sigma\mathcal{Z})_o^{\otimes p_i}(t,z) Q_i(t,z) \exp(D_{i,2}(t,z))}\vert < R$$

$$\textnormal{ for every }i \in \{1,...,s\}\}.$$

Claim 3

There is a regular $\kappa$-persistent $M^+: \pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft $N^+: \pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},N^+,M^+) \subset \Lambda_g \cap Y$.

Proof of Claim 3

For $i \in \{1,...,s\}$ let

$$\phi_i: Y \to \mathbb{C}, (t,z) \mapsto A_i(t)(\sigma\mathcal{Z})_o^{\otimes p_i}(t,z)Q_i(t,z)\exp(D_{i,2}(t,z)).$$

It is enough to show that there is a regular $\kappa$-persistent $M_i^+: \pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft $N_i^+: \pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},N_i^+,M_i^+) \subset \mathcal{B}(\phi_i,R) \cap Y$. As in the proof of Claim 1 we consider two cases.

Case 1:

Let $(p_{i\kappa},...,p_{ir}) \neq 0$. Consider the regular

$$U:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \left\{\begin{array}{ll} \frac{\sqrt{R}}{\vert{A_i(t)}\vert}, & t \in \pi(Y), \\
0, & \textnormal{ else.} \end{array}\right.$$

Since $A_i$ is consistent we see that $U_{\textnormal{pers}}$ is also consistent. So with the proof of Subclaim 1 in Claim 1 we obtain that $U$ is $\kappa$-persistent. For $(t,z) \in \mathcal{B}((\sigma\mathcal{Z})^{\otimes p_i}_o,U) \cap Y$ we have $\vert{\phi_i(t,z)}\vert<R$ since $\vert{Q_i(t,z)\exp(D_{i,2}(t,z))}\vert < \sqrt{R}$.

Suppose that $p_i$ is $\kappa$-positive. By Proposition 5.81(1) if $p_{i \kappa} \neq 0$ respectively Proposition 5.81(2) if $\kappa=k \geq 0$ and $p_{i \kappa} = 0$ (since $U_{\textnormal{pers}}$ is consistent) there is a regular $\kappa$-persistent $\hat{M}_i^+:\pi(C) \to \mathbb{R}_{ \geq 0}$ such that $\mathcal{B}(\mathcal{P}_{0,\kappa},\hat{M}_i^+) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes p_i},U)$. Choose $N_i^+:=S_\delta$ and

$$M_i^+:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \min\{\hat{M}_i^+(t),T_\delta(t)\}.$$

Then $\mathcal{A}(\mathcal{P}_{0,\kappa},N_i^+,M_i^+) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes p_i},U) \cap Y$.

Suppose that $p_i$ is $\kappa$-negative. Note that $U_{\textnormal{soft}}$ is $\kappa$-soft (since $\vert{b_i}\vert$ is $\kappa$-soft by Proposition 5.44(1), compare also with the proof of Subclaim 1 in Claim 1). With Corollary 5.82 we find a regular $\kappa$-soft $\hat{N}_i^+:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $\hat{M}_i^+:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $\mathcal{A}(\mathcal{P}_{0,\kappa},\hat{N}_i^+,\hat{M}_i^+) \subset \mathcal{B}((\sigma\mathcal{Z})^{\otimes p_i},U)$. The rest of the proof is similar as in the ”$\kappa$-positive”-case.

Case 2:

Let $(p_{i\kappa},...,p_{ir}) = 0$. Take $N_i^+:=S_\delta$.

Note that if $\vert{A_i(t)}\vert<\sqrt{R}$ then

$$\vert{\phi_i(t,z)}\vert = \vert{A_i(t)Q_i(t,z)\exp(D_{i,2}(t,z))\vert}<\sqrt{R} \cdot \sqrt[4]{R} \cdot \sqrt[4]{R}=R$$

for $(t,z) \in Y$ by (++) and (+++) (since $T_\delta \leq T_\epsilon$).  Let

$$M_i^+:\pi(C) \to \mathbb{R}_{\geq 0}, t \mapsto \left\{\begin{array}{ll} T_\delta(t), & t \in \pi(Y) \textnormal{ and } \vert{A_i(t)}\vert<\sqrt{R}, \\
0, & \textnormal{ else.} \end{array}\right.$$

Note that $M_i^+$ is regular and by construction that $\mathcal{A}(\mathcal{P}_{0,\kappa},N_i^+,M_i^+) \subset \mathcal{B}(\phi_i,R) \cap Y$. We show that $M_i^+$ is $\kappa$-persistent and are done.

Note that

$$\vert{A_i(t)Q_i(t,z)\exp(D_{i,2}(t,z))}\vert \leq 1$$

for every $(t,z) \in Y \cap C$. Let $w \in \pi_l(C)$. Let $\gamma:\textnormal{}]0,1[\textnormal{} \to C_w$ be a definable curve compatible with $C_w$ with $\lim_{y \searrow 0} (\gamma(y)) \in X_w$ and

$$\lim_{y \searrow 0} (\gamma_x(y)-\mu_\kappa(w,\gamma_u(y)))=0.$$

By passing to a suitable subcurve of $\gamma$ if necessary we may assume that $(w,\gamma_u(y)) \in \pi(Y)$ for $y \in \textnormal{}]0,1[$ (compare with the proof of Subclaim 1 in Claim 1) and that there is a definable continuous $\hat{\gamma}_x: \textnormal{}]0,1[\textnormal{} \to \mathbb{R}$ with $(\gamma_u(y),\hat{\gamma}_x(y)) \in C_w \cap Y_w$ for $y \in \textnormal{}]0,1[$. We have

$$\vert{A(w,\gamma_u(y))Q_i(w,\gamma_u(y),\gamma_x(y))e^{D_{i2}(w,\gamma_u(y),\gamma_x(y))}}\vert \leq 1$$

and therefore

\begin{eqnarray*}
\vert{A(w,\gamma_u(y))}\vert  \leq [Q_i(w,\gamma_u(y),\gamma_x(y))e^{D_{i2}(w,\gamma_u(y),\gamma_x(y))}]^{-1} \leq \sqrt{R}
\end{eqnarray*}

for $y \in \textnormal{}]0,1[$. So we have

$$\lim_{y \searrow 0} M_i^+(w,\gamma_u(y)) = \lim_{y \searrow 0} T_\delta(w,\gamma_u(y))>0.$$

This gives Claim 3.
$\blacksquare_{\textnormal{Claim }3}$

Take $N:=N^+$ and $M:=M^+$ where $N^+$ and $M^+$ are from Claim 3. From now on we consider $A_i$ as a function on $\mathcal{A}(\mathcal{P}_{0,\kappa},N(t),M(t))$ for $i \in \{0,...,s\}$. We obtain the following:

If $p_i$ is $\kappa$-negative for $i \in \{1,...,s\}$ then $1/((\vert{A_i}\vert^C)_{\textnormal{soft}})$ is $\kappa$-soft (since $\vert{b_i}\vert$ is $\kappa$-soft if $p_i$ is $\kappa$-negative by Proposition 5.44(1)) and if $g \in \log(E)$ and $q$ is $\kappa$-negative then $1/((\vert{A_0}\vert^C)_{\textnormal{soft}})$ is $\kappa$-soft  (since by Proposition 5.44(2) $\vert{a}\vert$ is $\kappa$-soft for $g \in \log(E)$ if $q$ is $\kappa$-negative) and we are done.

Outgoing from this preparation result we are able to prove our desired result on integration for a holomorphic extension of a prepared restricted log-exp-analytic function in one variable.

5.85 Corollary

There is a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ such that the following holds. Let

$$\Delta:=\{(t,s,z) \in \pi(C) \times \mathbb{R}_{>0} \times\mathbb{C} \textnormal{ }\mid N(t) < s < M(t),z \in B(\mu_{\kappa}(t),s)\}.$$

Then the function

$$\Psi:\Delta \to \mathbb{C}, (t,s,z) \mapsto \frac{1}{2\pi i} \int_{\partial B(\mu_\kappa(t),s)} \frac{F(t,\xi)}{\xi-z}d\xi,$$

is well-defined. Additionally there is $l \in \mathbb{N}$ and a definable function $g = (g_1,...,g_l):\pi^+(\Delta) \to \mathbb{R}^l$ such that $\Psi$ is constructible in $z$ with support function

$$\pi^+(\Delta) \to \mathbb{R}^l \times \mathbb{R} \times \mathbb{R}, (t,s) \mapsto (g(t,s),\mu_\kappa(t),s),$$

where the following holds for $g_j$ for $j \in \{1,...,l\}$: There is $p \in \mathbb{N}_0$, a regular function $\beta: \pi(C) \to \mathbb{R}^p$ and a log-analytic $h_j:\mathbb{R}^{p+1} \to \mathbb{R}$ such that $g_j(t,s)=h_j(\beta(t),s)$ for every $(t,s) \in \pi^+(\Delta)$.

By Proposition 5.84 there is a regular $\kappa$-persistent $\hat{M}:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $F|_{\hat{W}}$ is complex $r$-log-analytically prepared in $z$ with center $\Theta$ with regular coefficient and base functions where $\hat{W}:=\mathcal{A}(\mathcal{P}_{0,\kappa},N,\hat{M})$. Let

be a complex LA-preparing tuple for $F|_{\hat{W}}$ where $a$ and $b$ are regular.

By Corollary 5.68(2) if $\kappa=r$ respectively Proposition 5.74 if $\kappa=k$ we find a regular $\kappa$-persistent $\tilde{M}:\pi(C) \to \mathbb{R}_{\geq 0}$ (with $\tilde{M}<\vert{\mu_r-\mu_k}\vert$ if $k \geq 0$) such that for $W:=\mathcal{A}(\mathcal{P}_{0,\kappa},N,\tilde{M})$ we have $W \subset \hat{W}$ and

$$z_\alpha^*:W(\mu_{\kappa}) \to \mathbb{C}, (t,s,\tau) \mapsto z_\alpha(t,\mu_\kappa(t)+(\prod_{j=0}^\kappa \sigma_j) se^{i \tau}),$$

is free-regular in $s$ for $\alpha \in \{0,...,r\}$. (Note that

$$W(\mu_\kappa)=\{(t,s) \in \pi(C) \times \mathbb{R}_{>0} \textnormal{ }\mid N(t) < s < \hat{M}(t)\} \times \textnormal{}]-\pi,\pi[.)$$

Let $\eta:=(a,b_1,...,b_s)$. Note that $\eta$ is regular. Similarly as in the proof of the claim in Remark 5.79 there is a globally subanalytic function $G:\mathbb{C}^{s+1} \times \mathbb{C}^{r+1} \to \mathbb{C}$ such that

for every $(t,z) \in W$. With Remark 5.66(2) we have that

$$F^*:W(\mu_\kappa) \to \mathbb{C}, (t,s,\tau) \mapsto F(t,\mu_\kappa(t)+(\prod_{j=0}^\kappa \sigma_j) se^{i \tau}),$$

is free-regular in $s$ (since the components of $\eta$ considered as functions on $W(\mu_\kappa)$ are free-regular in $s$). By shrinking $\tilde{M}$ if necessary we may assume that $F^*$ is bounded in $\tau$. (By Corollary 5.69 if $\kappa=r$ respectively Corollary 5.75 if $\kappa=k$ we obtain after shrinking $\tilde{M}$ if necessary that

$$W(\mu_\kappa) \to \mathbb{C}, (t,s,\tau) \mapsto (\sigma\mathcal{Z})^{\otimes q}(t,\mu_\kappa(t)+(\prod_{j=0}^\kappa \sigma_j) se^{i \tau}),$$

is bounded in $\tau$.) Choose $M:=\tilde{M}$. Let $l \in \mathbb{N}$ and $g:=(g_1,...,g_l):\pi^+(W(\mu_\kappa)) \to \mathbb{R}^l$ be a corresponding support function for $F^*$. With Proposition 5.63 we see that the function

$$\Psi:\Delta \to \mathbb{C}, (t,s,z) \mapsto \frac{1}{2 \pi i}\int_{\partial B(\mu_{\kappa}(t),s)}\frac{F(t,\xi)}{\xi-z} d\xi,$$

is well-defined and constructible in $z$ with support function

$$\pi^+(\Delta) \to \mathbb{R}^l \times \mathbb{R} \times \mathbb{R}, (t,s) \mapsto (g(t,s),\mu_\kappa(t),s).$$

Since $\pi^+(W(\mu_\kappa))=\pi^+(\Delta)$ we see with Definition 5.64 that $g_j$ fulfills the desired properties for $j \in \{1,...,l\}$.

We can also strengthen Proposition 5.84 in the following way.

5.86 Proposition

There is a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft
$N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that the following holds where $W:=\mathcal{A}(\mathcal{P}_{0,\kappa},N,M)$.

(1) $F|_W$ is complex $r$-log-analytically prepared as in Proposition 5.84.

(2) If $N(t)=0$, $M(t)>0$ and $F(t,-)$ has a holomorphic extension at $\mu:=\mu_\kappa(t)$ (i.e. there is an open neighbourhood $U$ of $\mu$ in $B(\mu,M(t))$ and a holomorphic $J:U \to \mathbb{C}$ such that $J|_{U \cap W_t} = F(t,-)$) then there is a holomorphic function $h: B(\mu,M(t)) \to \mathbb{C}$ such that $h|_{W_t}=F(t,-)$.

By Proposition 5.84 there is a regular $\kappa$-persistent $M:\pi(C) \to \mathbb{R}_{\geq 0}$ and a regular $\kappa$-soft $N:\pi(C) \to \mathbb{R}_{\geq 0}$ such that $F|_{\mathcal{A}(\mathcal{P}_{0,\kappa}N,M)}$ is complex $r$-log-analytically prepared in $z$ with complex LA-preparing tuple

such that $q_0=...=q_{\kappa-1}=0$, $P \in M_\kappa(s \times r+1, \mathbb{Q})$, $a$ and $b$ are regular. By shrinking $M$ if necessary we can suppose

$$\bigl|\frac{\mathcal{P}_{j,\kappa}(t,z)}{e^{\mu_{\kappa-j-1,\kappa}(t)}} \bigl| < 1/2$$

for $(t,z) \in D$ and $j \in \{0,...,\kappa-1\}$ (compare with Proposition 5.67(1)). $(+)$

We have

$$F(t,-)=a(t)(\sigma\mathcal{Z})^{\otimes q}(t,z)V(b_1(t)(\sigma\mathcal{Z})^{\otimes p_1}(t,z),...,b_s(t) (\sigma\mathcal{Z})^{\otimes p_s}(t,z)).$$

Since $z_\kappa(t,z)=\mathcal{P}_\kappa(t,z)$ we obtain with $(+)$ by successively using the logarithmic series

$$z_\kappa(t,z)=L(z-\mu) \textnormal{ } (*)$$

for $z \in D_t$ where $L$ is a power series around zero which converges absolutely on $B(0,M(t))$.

We set $Z:=(Z_1,...,Z_s)$. Let $\sum_{\alpha \in \mathbb{N}_0^s}c_\alpha Z^{\alpha}$ be the power series expansion of $V$.

We may assume that $a(t) \neq 0$. Note that $(p_{j\kappa},...,p_{jr}) = 0$ or $p_j$ is $\kappa$-positive for $j \in \{1,...,s\}$. Let

$$\Gamma_1:=\{\alpha \in \mathbb{N}_0^s \textnormal{ } \big\vert\;^tP\alpha+q \in \{0\}^\kappa \times \mathbb{N}_0 \times \{0\}^{r-\kappa}\}$$

and $\Gamma_2:=\mathbb{N}_0^s \setminus \Gamma_1$. Set $V_1:=\sum_{\alpha \in \Gamma_1}c_\alpha Z^\alpha$ and $V_2:=\sum_{\alpha \in \Gamma_2} c_\alpha Z^{\alpha}$. For $l \in \{1,2\}$ let

$$G_l:D_t \to \mathbb{C}, z \mapsto a(t)(\sigma\mathcal{Z})^{\otimes q}(t,z)V_l(b_1(t)(\sigma\mathcal{Z})^{\otimes p_1}(t,z),...,b_s(t)(\sigma\mathcal{Z})^{\otimes p_s}(t,z)).$$

So we see by composition of power series and $(*)$ that

$$G_1(z)=\sum_{j=0}^{\infty} d_j(z-\mu)^j$$

for $z \in D_t$ where $d_j \in \mathbb{C}$ and the series to the right converges absolutely on $B(\mu,M(t))$. So $G_1$ has a holomorphic extension $\hat{G}_1$ on $B(\mu,M(t))$. Let $0<\epsilon<M(t)$ be such that $F|_{B(\mu,\epsilon) \cap D_t}$ has a holomorphic extension $\hat{F}$ on $W:=B(\mu,\epsilon)$. Set

$$\hat{G}_2:W \to \mathbb{C}, z \mapsto \hat{F}(z)-\hat{G}_1(z).$$

Note that $\hat{G}_2$ is holomorphic and that $\hat{G}_2|_{D_t \cap W}=G_2$. We show that $\hat{G}_2=0$ and we are done by the identity theorem (since $F_t$ coincides then with $G_1$ on $W \cap D_t$). Let

$$\Omega:=\{(\omega_\kappa,...,\omega_r) \in \mathbb{Q}^{r-\kappa+1} \mid \textnormal{ there is }\alpha \in \Gamma_2 \textnormal{ such that }$$

$$(^t P \alpha + q)_j=\omega_j \textnormal{ for } j \in \{\kappa,...,r\}\}.$$

Then $\Omega \subset \mathbb{Q}^{r-\kappa+1} \setminus (\mathbb{N}_0 \times \{0\}^{r-\kappa})$. For $\omega \in \Omega$ let

$$\Gamma_{2,\omega}:=\{\alpha \in \mathbb{N}_0^s \mid (^tP\alpha + q)_j = \omega_j \textnormal{ for } j \in \{\kappa,...,r\}\}.$$

We have

$$\hat{G}_2(z)=\sum_{\omega \in \Omega}e_\omega \prod_{j=\kappa}^r (\sigma_jz_j(t,z))^{\omega_j}$$

$$e_\omega=a(t)\sum_{\alpha \in \Gamma_{2,\omega}}c_\alpha \prod_{j=1}^s b_j(t)^{\alpha_j}$$

for $z \in W$. Let

$$\Omega^*:=\{\omega \in \Omega \mid e_{\omega} \neq 0 \}.$$

If $\Omega^* = \emptyset$ we are done. So assume $\Omega^* \neq \emptyset$.
There is $\xi:=(\xi_\kappa,...,\xi_r) \in \Omega$ such that $\prod_{j=\kappa}^r \vert{z_j}\vert^{\omega_j} = o(\prod_{j=\kappa}^r \vert{z_j}\vert^{\xi_j})$
for all $\omega \in \Omega$ with $\omega \neq \xi$. By investigating the following two cases we are done.

Case 1:

If $\xi_\kappa \in \mathbb{N}_0$ (note that $\xi \neq 0$) we differentiate $\hat{G}_2$ $m$-times with respect to $z$ where $m:=\xi_\kappa+1$ and we see with Corollary 5.35 (note that $\xi \neq 0$) that there is $\beta=(-1,\beta_{\kappa+1},...,\beta_r) \in \mathbb{Q}^{r-\kappa+1}$ such that

$$\lim_{z \to \mu} \Bigl|\frac{d^m \hat{G}_2/dz^m(z)}{\prod_{j=\kappa}^r z_j^{\beta_j}} \Bigl| \in \mathbb{R}^*.$$

But we obtain