$\Phi_{b}\mathopen{}\mathclose{{}\left(z}\right)$ is that it is invariant
under inversion of $b$ with $z$ kept fixed,

$\Phi_{b}\mathopen{}\mathclose{{}\left(z}\right)=\Phi_{1/b}\mathopen{}\mathclose{{}\left(z}\right)\,.$
(A.176)

In a physics language this corresponds to a sort of S-duality292929See also
Wang:2015wdy ; Hatsuda:2015qzx  for a discussion about the
$\hbar\leftrightarrow 1/\hbar$ symmetry in the context of the TS/ST
correspondence and relativistic integrable systems. since it inverts the
string coupling $g_{s}$.

An integral representation of the quantum dilogarithm which will be useful for
us is (Hatsuda:2016uqa, , eq. A.2)

$\begin{split}\log\mathopen{}\mathclose{{}\left(\Phi_{b}\mathopen{}\mathclose{{}\left(z}\right)}\right)&=-\frac{\mathrm{i}}{2\pi
b^{2}}\mathrm{Li}_{2}\mathopen{}\mathclose{{}\left(-\mathrm{e}^{2\pi
bz}}\right)-\mathrm{i}\int_{0}^{+\infty}\frac{\mathrm{d}t}{1+\mathrm{e}^{2\pi
t}}\log\mathopen{}\mathclose{{}\left(\frac{1+\mathrm{e}^{2\pi
bz}\mathrm{e}^{-2\pi b^{2}t}}{1+\mathrm{e}^{2\pi bz}\mathrm{e}^{2\pi
b^{2}t}}}\right)\\\
&=-\frac{\mathrm{i}}{2\pi}b^{2}\mathrm{Li}_{2}\mathopen{}\mathclose{{}\left(-\mathrm{e}^{2\pi
b^{-1}z}}\right)-\mathrm{i}\int_{0}^{+\infty}\frac{\mathrm{d}t}{1+\mathrm{e}^{2\pi
t}}\log\mathopen{}\mathclose{{}\left(\frac{1+\mathrm{e}^{2\pi
b^{-1}z}\mathrm{e}^{-2\pi b^{-2}t}}{1+\mathrm{e}^{2\pi
b^{-1}z}\mathrm{e}^{2\pi b^{-2}t}}}\right),\end{split}$ (A.177)

where the second representation follows from the inversion symmetry of the
quantum dilogarithm (A.176). It is this second integral representation which
will be useful in taking the 4d limit.

###  A.2 A four-dimensional limit

Consider the following reparametrization of the Kähler parameters (bgt, , p.
5)

$X=\exp\mathopen{}\mathclose{{}\left(g_{s}x}\right)\,,\qquad\qquad
Q_{B}=g_{s}^{4}t\,,\qquad\qquad
Q_{F}=\exp\mathopen{}\mathclose{{}\left(2g_{s}\mathrm{i}\sigma}\right)\,,$
(A.178)

where as before $q=\exp\mathopen{}\mathclose{{}\left(\mathrm{i}g_{s}}\right)$
and
$q^{S}=\exp\mathopen{}\mathclose{{}\left(\mathrm{i}\mathopen{}\mathclose{{}\left(2\pi}\right)^{2}/g_{s}}\right)$.
The four-dimensional limit is then $g_{s}\to 0$ with $x$, $t$ and $\sigma$ and
kept fixed. In term of more familiar gauge theoretic quantities one has

$x=\frac{y}{2\epsilon}\,,\qquad\qquad
t=\mathopen{}\mathclose{{}\left(\frac{\Lambda}{\epsilon}}\right)^{4}\,,\qquad\qquad\sigma=\mathrm{i}\frac{a}{2\epsilon}\,,$
(A.179)

where $\epsilon$ corresponds to the $\Omega$-background parameter in the self-
dual phase, $\Lambda$ is the instanton counting parameter and $a$ is the
Coulomb branch parameter or equivalently the A-period of the underlying
Seiberg-Witten geometry. The parameter $y$ corresponds to the position of the
defect. Note also that the variable $x$ is called $q$ in the main text.

The instanton part of the type II defect is obtained as the $g_{s}\to 0$ limit
of the GV instanton part (A.170) of the topological string partition function
in the parametrization (A.178),

$Z_{\rm
inst}^{\text{II}}\mathopen{}\mathclose{{}\left(x,t,\sigma}\right)=\lim_{g_{s}\to
0}Z_{\mathbb{F}_{0}}^{\text{GV,
inst}}\mathopen{}\mathclose{{}\left(\exp\mathopen{}\mathclose{{}\left(g_{s}x}\right),g_{s}^{4}t,\exp\mathopen{}\mathclose{{}\left(2g_{s}{\rm
i}\sigma}\right),g_{s}}\right)=1-\mathopen{}\mathclose{{}\left[\frac{\widetilde{x}}{2\sigma^{2}\mathopen{}\mathclose{{}\left(\widetilde{x}^{2}-\sigma^{2}}\right)}}\right]t\\\
+\mathopen{}\mathclose{{}\left[\frac{\widetilde{x}\mathopen{}\mathclose{{}\left(\widetilde{x}+1}\right)^{2}-\widetilde{x}\mathopen{}\mathclose{{}\left(10\widetilde{x}^{2}+19\widetilde{x}+10}\right)\sigma^{2}+\mathopen{}\mathclose{{}\left(8\widetilde{x}^{2}+30\widetilde{x}+9}\right)\sigma^{4}}{4\sigma^{4}\mathopen{}\mathclose{{}\left(4\sigma^{2}-1}\right)^{2}\mathopen{}\mathclose{{}\left(\widetilde{x}^{2}-\sigma^{2}}\right)\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\widetilde{x}+1}\right)^{2}-\sigma^{2}}\right)}}\right]t^{2}+\mathcal{O}\mathopen{}\mathclose{{}\left(t^{3}}\right)~{},$
(A.180)

where we used for the sake of readability

$\widetilde{x}=\mathrm{i}x+\frac{1}{2}~{}.$ (A.181)

We checked (A.180) against (Sciarappa:2017hds, , eqs A.3-A.13) up to and
including order $t^{2}$, and found perfect agreement303030Looking at the
product of the instanton parts of the Nekrasov (3.18) and defect (6.110)
(A.180) partition functions, one finds equality with the “chiral defect”
(Sciarappa:2017hds, , eq A.7) if $\sigma_{\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Sciarappa:2017hds}{\@@citephrase{(}}{\@@citephrase{)}}}}}=-\epsilon\mathopen{}\mathclose{{}\left(\widetilde{x}-1}\right)$,
and with the “anti-chiral” defect (Sciarappa:2017hds, , eq A.10) if
$\sigma_{\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Sciarappa:2017hds}{\@@citephrase{(}}{\@@citephrase{)}}}}}=\epsilon\mathopen{}\mathclose{{}\left(\widetilde{x}-1}\right)$.
Note that since we are working in the self-dual phase of the
$\Omega$-background we have $\epsilon=\epsilon_{1}=-\epsilon_{2}$.. On the
other hand, one can see that the NS instanton part in (A.170) vanishes in the
same four-dimensional limit.

In the 4d limit the polynomial part (A.169) reduces to

$Z^{\text{II,(p)}}\mathopen{}\mathclose{{}\left(x}\right)=\lim_{g_{s}\to
0}Z^{\text{(p)}}\mathopen{}\mathclose{{}\left(\exp\mathopen{}\mathclose{{}\left(g_{s}x}\right),g_{s}}\right)=\exp\mathopen{}\mathclose{{}\left(-\pi
x}\right)=\exp\mathopen{}\mathclose{{}\left(-\pi\frac{y}{2\epsilon}}\right).$
(A.182)

Looking at the 1-loop part (A.174) and (A.175) and the reparametrization
(A.178) one can see that

$-\mathrm{e}^{2\pi
b^{-1}z_{\pm}}=\mathrm{e}^{-g_{s}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)},$ (A.183)

and using the second integral representation of Faddeev’s quantum dilogarithm
(A.177) gives

$\log\mathopen{}\mathclose{{}\left(\Phi_{b}\mathopen{}\mathclose{{}\left(z_{\pm}}\right)}\right)=-\mathrm{i}\frac{\mathrm{Li}_{2}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-g_{s}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)}}\right)}{g_{s}}-\mathrm{i}\int_{0}^{+\infty}\frac{\mathrm{d}t}{1+\mathrm{e}^{2\pi
t}}\log\mathopen{}\mathclose{{}\left(\frac{1-\mathrm{e}^{-g_{s}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)}\mathrm{e}^{-g_{s}t}}{1-\mathrm{e}^{-g_{s}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)}\mathrm{e}^{g_{s}t}}}\right).$ (A.184)

The expansion of for $g_{s}\to 0$ from above

$-\mathrm{i}\frac{\mathrm{Li}_{2}\mathopen{}\mathclose{{}\left(\mathrm{e}^{-g_{s}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)}}\right)}{g_{s}}=-\mathrm{i}\frac{\pi^{2}}{6}\frac{1}{g_{s}}-\mathrm{i}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)\log\mathopen{}\mathclose{{}\left(g_{s}}\right)-\mathrm{i}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)\mathopen{}\mathclose{{}\left[\log\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)-1}\right]+\mathcal{O}\mathopen{}\mathclose{{}\left(g_{s}}\right),$
(A.185) $-\mathrm{i}\int_{0}^{+\infty}\frac{\mathrm{d}t}{1+\mathrm{e}^{2\pi
t}}\log\mathopen{}\mathclose{{}\left(\frac{1-\mathrm{e}^{-g_{s}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)}\mathrm{e}^{-g_{s}t}}{1-\mathrm{e}^{-g_{s}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)}\mathrm{e}^{g_{s}t}}}\right)=-\mathrm{i}\int_{0}^{+\infty}\frac{\mathrm{d}t}{1+\mathrm{e}^{2\pi
t}}\log\mathopen{}\mathclose{{}\left(\frac{x\pm{\rm i}\sigma+t}{x\pm{\rm
i}\sigma-t}}\right)+\mathcal{O}\mathopen{}\mathclose{{}\left(g_{s}}\right),$
(A.186)

The divergent terms in (A.185) can be dealt with properly in the context of
the open TS/ST correspondence matijn-alba . For the propose of this work we
will simply drop them. By using the integral representation of the Gamma
function we get (adamchik2003contributions, , p. 8)

$-\mathrm{i}\int_{0}^{+\infty}\frac{\mathrm{d}t}{1+\mathrm{e}^{2\pi
t}}\log\mathopen{}\mathclose{{}\left(\frac{x\pm{\rm i}\sigma+t}{x\pm{\rm
i}\sigma-t}}\right)=-2\int_{0}^{+\infty}\frac{\mathrm{d}t}{1+\mathrm{e}^{2\pi
t}}\arctan\mathopen{}\mathclose{{}\left(\frac{t}{-\mathrm{i}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)}}\right)\\\ =\mathrm{i}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)\log\mathopen{}\mathclose{{}\left(-\mathrm{i}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)}\right)-\mathrm{i}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)-\frac{\log\mathopen{}\mathclose{{}\left(2\pi}\right)}{2}+\log\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)+\frac{1}{2}}\right),$ (A.187)

where $\log\Gamma$ is the log Gamma function. Hence we get

$\lim_{g_{s}\to
0}\log\mathopen{}\mathclose{{}\left(\Phi_{b}\mathopen{}\mathclose{{}\left(z_{\pm}}\right)}\right)=\log\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)+\frac{1}{2}}\right)-\frac{\log\mathopen{}\mathclose{{}\left(2\pi}\right)}{2}+\frac{\pi}{2}\mathopen{}\mathclose{{}\left(x\pm{\rm
i}\sigma}\right)$ (A.188)

and combining with (A.182) gives the perturbative part of the type II defect
partition function

$\begin{split}Z^{\text{II}}_{\rm
pert}\mathopen{}\mathclose{{}\left(x,\sigma}\right)&=\lim_{g_{s}\to
0}Z_{\mathbb{F}_{0}}^{\text{(p)}}\mathopen{}\mathclose{{}\left(\exp\mathopen{}\mathclose{{}\left(g_{s}x}\right),g_{s}}\right)Z_{\mathbb{F}_{0}}^{\text{1-loop}}\mathopen{}\mathclose{{}\left(\exp\mathopen{}\mathclose{{}\left(g_{s}x}\right),\exp\mathopen{}\mathclose{{}\left(2g_{s}{\rm
i}\sigma}\right),g_{s}}\right)\\\
&=\frac{\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}\mathopen{}\mathclose{{}\left(x-{\rm
i}\sigma}\right)+\frac{1}{2}}\right)\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}\mathopen{}\mathclose{{}\left(x+{\rm
i}\sigma}\right)+\frac{1}{2}}\right)}{2\pi}.\end{split}$ (A.189)

The product of the pertubative (A.189) and instanton (A.180) (A.181) parts
give us then the complete partition function for the type II defect in 4d,
$\mathcal{N}=2$, $\mathrm{SU}\mathopen{}\mathclose{{}\left(2}\right)$ super
Yang-Mills313131We choose for the four-dimensional partition function a
normalization that does not include the $1/2\pi$. In addition the relation
between eigenfunctions of the integral kernel (3.14) and the defect partition
function (A.190) makes it convenient to include an extra factor
$\exp\mathopen{}\mathclose{{}\left(\mathrm{i}x\log\mathopen{}\mathclose{{}\left(t}\right)/2}\right)$
in the latter.,

$Z^{\text{II}}\mathopen{}\mathclose{{}\left(x,t,\sigma}\right)=\exp\mathopen{}\mathclose{{}\left(\frac{\mathrm{i}}{2}x\log\mathopen{}\mathclose{{}\left(t}\right)}\right)\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}x-\sigma+\frac{1}{2}}\right)\Gamma\mathopen{}\mathclose{{}\left(-\mathrm{i}x+\sigma+\frac{1}{2}}\right)Z_{\rm
inst}^{\text{II}}\mathopen{}\mathclose{{}\left(x,t,\sigma}\right)~{}.$ (A.190)

Note that we use a slightly different notation compared to the main text: what
we call $x$ here is called $q$ elsewhere.

##  Appendix B From the matrix model identity to the eigenfunction identity

In this appendix we argue for the equivalence between the identities (6.114)
and (6.115)323232 We will not be rigorous in switching the order of sums and
integrals since all the quantities are conjecturally convergent.. Our strategy
is similar to the one used in the context of ABJM theory, see e.g.
Hatsuda:2012dt  and reference therein.

Let us start by writing (6.114) as

$\int_{{\mathbb{R}}+{\rm i}\sigma_{*}}{{\rm
d}\sigma}{\tan\mathopen{}\mathclose{{}\left(2\pi\sigma}\right)\over\mathopen{}\mathclose{{}\left(2\cos(2\pi\sigma)}\right)^{N}}\mathopen{}\mathclose{{}\left(Z^{\rm
II}_{\rm tot}\mathopen{}\mathclose{{}\left(q,t,\sigma}\right)+Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over 2},t,\sigma+{1\over
2}}\right)}\right)\\\ =\mathrm{i}{2^{11/12}\sqrt{\pi}{t^{3/16}}\over{\rm
e}^{3\zeta^{\prime}(-1)}{\rm e}^{4\sqrt{t}}(4\pi)^{N}}\int_{\mathbb{R}}{{\rm
d}x}\ {\rm e}^{-{\rm i}{2}qx}{\rm e}^{-4t^{1/4}\cosh x+{x\over
2}}\Psi_{N}({\rm e}^{{x}},t),$ (B.191)

where we have absorbed the $(-1)^{N}$ into a shift of $\sigma$, and
$\sigma_{*}$ is a strictly positive number which guarantees that the
integration contour on the left hand side of (B.191) does not hit the poles of
$Z^{\rm II}_{\rm tot}$. This is the case if $0<\sigma_{*}<|{\rm Re}(q)|\neq
0.$ If ${\rm Re}(q)=0$ one can take $\sigma_{*}$ to be any strictly positive
number as in footnote 11. For the sake of notation let us define

$\displaystyle f(N)$ $\displaystyle={\rm
i}{2^{11/12}\sqrt{\pi}{t^{3/16}}\over{\rm e}^{3\zeta^{\prime}(-1)}{\rm
e}^{4\sqrt{t}}(4\pi)^{N}}\int_{\mathbb{R}}{{\rm d}x}\ {\rm e}^{-{\rm
i}{2}qx}{\rm e}^{-4t^{1/4}\cosh x+{x\over 2}}\Psi_{N}({\rm e}^{{x}},t)\,,$
(B.192) $\displaystyle g(\sigma)$ $\displaystyle=Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(q,t,\sigma}\right)+Z^{\rm II}_{\rm
tot}\mathopen{}\mathclose{{}\left(-q-{{\rm i}\over 2},t,\sigma+{1\over
2}}\right)\,,$ (B.193)

so that (B.191) becomes

$f\mathopen{}\mathclose{{}\left(N}\right)=\int_{\mathbb{R}}\mathrm{d}\sigma\frac{\tan\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)}\right)}{\mathopen{}\mathclose{{}\left[2\cos\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)}\right)}\right]^{N}}g\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)\\\
=\int_{-1/2}^{1/2}\mathrm{d}\sigma\frac{\tan\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)}\right)}{\mathopen{}\mathclose{{}\left[2\cos\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)}\right)}\right]^{N}}\sum_{k\in\mathbb{Z}}g\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}+k}\right)~{}.$
(B.194)

Note that the second equality in (B.194) assumes some good analytic properties
of $g$, for instance $g$ is such that the sum over $n$ on the right hand side
of (B.194) is convergent. This is the case for (B.192). Furthermore it is part
of our conjecture that the function
$\sum_{k\in\mathbb{Z}}g\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}+k}\right)$
is not only well-defined but also an entire function of $\sigma$. Hence we are
free to deform the integration path in (B.194) to any path
$\mathcal{C}_{\mathopen{}\mathclose{{}\left\\{-1/2,1/2}\right\\}}$, beginning
at $\sigma=-1/2$ and ending at $\sigma=1/2$, as long as we don’t cross the
poles coming from the tangent when
$\sigma+\mathrm{i}\sigma_{*}\in\mathbb{Z}/2+1/2$. Consider then the change of
variables given by

$\mu:\mathcal{C}_{\mathopen{}\mathclose{{}\left\\{-\frac{1}{2},\frac{1}{2}}\right\\}}\to\mathopen{}\mathclose{{}\left]-\pi,\pi}\right[:\sigma\mapsto\mu\mathopen{}\mathclose{{}\left(\sigma}\right)=-\mathrm{i}\log\mathopen{}\mathclose{{}\left[\frac{\cos\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)}\right)}{2\pi}}\right],$
(B.195)

which is well-defined for a suitable choice of
$\mathcal{C}_{\mathopen{}\mathclose{{}\left\\{-1/2,1/2}\right\\}}$. Then we
have for the integral that

$\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}\mu\exp\mathopen{}\mathclose{{}\left(-\mathrm{i}\mu
N}\right)\cdots=-\mathrm{i}\mathopen{}\mathclose{{}\left(2\pi}\right)^{N}\int_{-1/2}^{1/2}\mathrm{d}\sigma\frac{\tan\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)}\right)}{\mathopen{}\mathclose{{}\left[\cos\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)}\right)}\right]^{N}}\cdots~{},$
(B.196)

and hence we can rewrite (B.194) as

$-\mathrm{i}\mathopen{}\mathclose{{}\left(4\pi}\right)^{N}f\mathopen{}\mathclose{{}\left(N}\right)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}\mu\exp\mathopen{}\mathclose{{}\left(-\mathrm{i}\mu
N}\right)\sum_{k\in\mathbb{Z}}g\mathopen{}\mathclose{{}\left(\sigma\mathopen{}\mathclose{{}\left(\mu}\right)+\mathrm{i}\sigma_{*}+k}\right),$
(B.197)

where $\sigma\mathopen{}\mathclose{{}\left(\mu}\right)$ is the inverse of
(B.195). Note that the integral on the right hand side gives the $N$-th
Fourier coefficient of the function
$\sum_{k\in\mathbb{Z}}g\mathopen{}\mathclose{{}\left(\sigma\mathopen{}\mathclose{{}\left(\mu}\right)+\mathrm{i}\sigma_{*}+k}\right)$.
The full Fourier series leads then to

$\begin{split}-\mathrm{i}\sum_{N\in\mathbb{N}}\mathopen{}\mathclose{{}\left(4\pi}\right)^{N}\exp\mathopen{}\mathclose{{}\left(\mathrm{i}\nu
N}\right)f\mathopen{}\mathclose{{}\left(N}\right)&=\sum_{N\in\mathbb{Z}}\frac{\exp\mathopen{}\mathclose{{}\left(\mathrm{i}\nu
N}\right)}{2\pi}\int_{-\pi}^{\pi}\mathrm{d}\mu\exp\mathopen{}\mathclose{{}\left(-\mathrm{i}\mu
N}\right)\sum_{k\in\mathbb{Z}}g\mathopen{}\mathclose{{}\left(\sigma\mathopen{}\mathclose{{}\left(\mu}\right)+\mathrm{i}\sigma_{*}+k}\right)\\\
&=\sum_{k\in\mathbb{Z}}g\mathopen{}\mathclose{{}\left(\sigma\mathopen{}\mathclose{{}\left(\nu}\right)+\mathrm{i}\sigma_{*}+k}\right).\end{split}$
(B.198)

where the last equality holds whenever the Fourier series on the previous line
is convergent. We expect this to be true in our case even tough we do not have
a rigorous proof. We also used that $f(-N)=0$ for
$N\in{\mathbb{N}}\setminus\mathopen{}\mathclose{{}\left\\{0}\right\\}$.

To go in the opposite direction from (B.198) to (B.194) one can apply Cauchy’s
residue theorem. Using the notation

$\kappa=\exp\mathopen{}\mathclose{{}\left(\mathrm{i}\nu}\right)=\frac{\cos\mathopen{}\mathclose{{}\left(2\pi\mathopen{}\mathclose{{}\left(\sigma+\mathrm{i}\sigma_{*}}\right)}\right)}{2\pi}$
(B.199)

one can multiply both sides in (B.198) by $1/\kappa^{N+1}$ and integrate over
$\kappa$ along a anticlockwise contour of radius 1 centered at the origin to
arrive at

$\begin{split}\mathopen{}\mathclose{{}\left(2\pi}\right)\mathopen{}\mathclose{{}\left(4\pi}\right)^{N}f\mathopen{}\mathclose{{}\left(N}\right)&=\oint\frac{\mathrm{d}\kappa}{\kappa^{N+1}}\sum_{k\in\mathbb{Z}}g\mathopen{}\mathclose{{}\left(\sigma\mathopen{}\mathclose{{}\left(-\mathrm{i}\log\mathopen{}\mathclose{{}\left(\kappa}\right)}\right)+\mathrm{i}\sigma_{*}+k}\right)\\\
&=\mathrm{i}\int_{-\pi}^{\pi}\mathrm{d}\nu\exp\mathopen{}\mathclose{{}\left(-\mathrm{i}\nu
N}\right)\sum_{k\in\mathbb{Z}}g\mathopen{}\mathclose{{}\left(\sigma\mathopen{}\mathclose{{}\left(\nu}\right)+\mathrm{i}\sigma_{*}+k}\right)\end{split}$
(B.200)

which is equivalent to (B.197) and hence by (B.196) to (B.194). We can
conclude that (B.194) and (B.198) are indeed equivalent, provided $f$ and $g$
have good analytic properties which we assume to be the case.

##  Appendix C Elliptic integrals

For the elliptic integrals we follow the conventions of Wolfram Mathematica
reference.wolfram_2023_elliptick ; reference.wolfram_2023_ellipticf ;
reference.wolfram_2023_elliptice ; reference.wolfram_2023_ellipticpi . The
notation in (byrd1971handbook, , pp. 8-10) is slightly different and we denote
their elliptic integrals with a tilde. In particular we have the normal or
incomplete elliptic integral of the first kind for $k^{2}\in\mathbb{R}$,
$-\pi/2<\phi<\pi/2$ and
$k^{2}\sin^{2}\mathopen{}\mathclose{{}\left(\phi}\right)<1$,
reference.wolfram_2023_ellipticf , (byrd1971handbook, , eq. 110.02),

$\mathrm{F}\mathopen{}\mathclose{{}\left(\phi\middle|k^{2}}\right)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-k^{2}\sin^{2}\mathopen{}\mathclose{{}\left(\theta}\right)}}=\widetilde{F}\mathopen{}\mathclose{{}\left(\phi,k}\right),$
(C.201)

the normal or incomplete elliptic integral of the second kind for $k^{2}<1$
and $-\pi/2<\phi<\pi/2$, reference.wolfram_2023_elliptice , (byrd1971handbook,
, eq. 110.03),

$\mathrm{E}\mathopen{}\mathclose{{}\left(\phi\middle|k^{2}}\right)=\int_{0}^{\phi}\mathrm{d}\theta\sqrt{1-k^{2}\sin^{2}\mathopen{}\mathclose{{}\left(\theta}\right)}=\widetilde{E}\mathopen{}\mathclose{{}\left(\phi,k}\right),$
(C.202)

and the normal or incomplete elliptic integral of the third kind for
$k^{2},\alpha^{2}\in\mathbb{R}$, $-\pi/2<\phi<\pi/2$ and
$k^{2}\sin^{2}\mathopen{}\mathclose{{}\left(\phi}\right)<1$,
reference.wolfram_2023_ellipticpi , (byrd1971handbook, , eq. 110.04),

$\Pi\mathopen{}\mathclose{{}\left(\alpha^{2};\phi\middle|k^{2}}\right)=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\mathopen{}\mathclose{{}\left(1-\alpha^{2}\sin^{2}\mathopen{}\mathclose{{}\left(\theta}\right)}\right)\sqrt{1-k^{2}\sin^{2}\mathopen{}\mathclose{{}\left(\theta}\right)}}=\widetilde{\Pi}\mathopen{}\mathclose{{}\left(\phi,\alpha^{2},k}\right).$
(C.203)

The complete elliptic integrals are obtained by taking $\phi=\pi/2$,

$\mathrm{K}\mathopen{}\mathclose{{}\left(k^{2}}\right)=\mathrm{F}\mathopen{}\mathclose{{}\left(\frac{\pi}{2}\middle|k^{2}}\right),\quad\qquad\mathrm{E}\mathopen{}\mathclose{{}\left(k^{2}}\right)=\mathrm{E}\mathopen{}\mathclose{{}\left(\frac{\pi}{2}\middle|k^{2}}\right),\quad\qquad\Pi\mathopen{}\mathclose{{}\left(\alpha^{2}\middle|k^{2}}\right)=\Pi\mathopen{}\mathclose{{}\left(\alpha^{2};\frac{\pi}{2}\middle|k^{2}}\right).$
(C.204)

The complete elliptic integrals of the first and second kind are analytic on
$\mathbb{C}$ apart from a branch cut along the positive real line for
$k^{2}\geqslant 1$, and the complete elliptic integral of the third kind is
analytic on $\mathbb{C}^{2}$ apart from similar branch cuts for
$k^{2},\alpha^{2}\geqslant 1$ reference.wolfram_2023_elliptick ;
reference.wolfram_2023_elliptice ; reference.wolfram_2023_ellipticpi .

We also need a Jacobi elliptic function $\mathrm{sn}$ which is an inverse of
the incomplete elliptic integral of the first kind (byrd1971handbook, , p.
18),

$\mathrm{sn}\mathopen{}\mathclose{{}\left(v\middle|k^{2}}\right)\,,\qquad\qquad\qquad\mathrm{sn}\mathopen{}\mathclose{{}\left(\mathrm{F}\mathopen{}\mathclose{{}\left(\phi\middle|k^{2}}\right)\middle|k^{2}}\right)=\sin\mathopen{}\mathclose{{}\left(\phi}\right)\,,$
(C.205)

which is sometimes called the sine amplitude.

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