m^{\partial},(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))\ell(x^{\prime},\xi^{\prime})\rangle.$

Now we compute the limite as ${\varepsilon}\to 0$ of
$\langle\tilde{m}_{1},\partial_{\xi_{d}}a(x^{\prime},0,\xi^{\prime},0)\rangle$.
We have

$\displaystyle\partial_{\xi_{d}}a(x^{\prime},x_{d},\xi^{\prime},\xi_{d})=\ell(x^{\prime},\xi^{\prime})\chi(\xi_{d}/{\varepsilon})\chi(x_{d}/{\varepsilon})+\ell(x^{\prime},\xi^{\prime})\chi^{\prime}(\xi_{d}/{\varepsilon})\chi(x_{d}/{\varepsilon})\xi_{d}/{\varepsilon}.$

Then
$\partial_{\xi_{d}}a(x^{\prime},0,\xi^{\prime},0)=\ell(x^{\prime},\xi^{\prime})$
and

$\displaystyle\langle\tilde{m}_{1}-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))m^{\partial},\partial_{\xi_{d}}a(x^{\prime},0,\xi^{\prime},0)\rangle$
$\displaystyle=\langle\tilde{m}_{1}-(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))m^{\partial},\ell(x^{\prime},\xi^{\prime})\rangle$
$\displaystyle=-\langle
m^{\partial},(\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime}))\ell(x^{\prime},\xi^{\prime})\rangle$

from (1.90) and (1.91). We deduce that $\tilde{m}_{1}=0$ and the last result
of the lemma from (1.89). ∎

###  A.4. Proof of Proposition 5.1

We prove the propagation result, first in interior which is a classical
result, second in a neighborhood of a point on the boundary. From previous
lemmas, it is a classical result. We give here the main argument to be
complete. The proof is consequence of Lemmas A.1—A.5.

####  A.4.1. Propagation in interior domain $\Omega$

From Proposition 5.2, we have $H_{p}m=0$. It is then classical that $m$ in
invariant by the flow of $H_{p}$. More precisely, let $\rho_{0}\in
T^{*}\Omega$ and we assume that $\gamma(s,\rho_{0})\in T^{*}\Omega$ for
$s\in[0,t]$. Let $a\in\mathscr{C}^{\infty}_{0}(\Omega\times\mathbb{R}^{d})$ be
such that $a(\gamma(-s,.))$ is supported in $T^{*}\Omega$ for $s\in[0,t]$, we
have $\langle\gamma_{*}(t,.)m,a\rangle=\langle
m,a(\gamma(-t,.))\rangle=\langle m,a\rangle$.

####  A.4.2. Propagation at boundary: hyperbolic points

We use the geometry context defined in section A.1, in particular the function
$j$ and the definition of the different flows.

We prove that the support of measure is locally empty in the future assuming
that in the past the support of measure is locally empty but by symmetry we
can deduce that the support of measure is locally empty in the past if we
assume that the support of measure is locally empty in the future.

Recall that we choose coordinates such that
$p(x,\xi)=\xi_{d}^{2}+R(x,\xi^{\prime})-1$ and locally $\Omega=\\{x_{d}>0\\}$.

We recall that a point $(x_{0}^{\prime},\xi_{0}^{\prime})\in
T^{*}\partial\Omega$ is in ${\mathcal{H}}$, if
$R(x_{0}^{\prime},0,\xi_{0}^{\prime})-1<0$. We apply Lemmas A.1 and A.3. We
have $m=1_{x_{d}>0}m$ and
$H_{p}m=m^{+}\otimes\delta_{\xi_{d}=\sqrt{1-R(x^{\prime},0,\xi^{\prime})}}-m^{+}\otimes\delta_{\xi_{d}=-\sqrt{1-R(x^{\prime},0,\xi^{\prime})}}$.

We call $\gamma^{\pm}$ the integral curve of $H_{p}$ starting from
$(x_{0}^{\prime},x_{d},\xi^{\prime}_{0},\pm\sqrt{1-R(x^{\prime},0,\xi^{\prime})})$.
If we assume that the support of $m=1_{x_{d}>0}m$ is empty in a neighborhood
of $\gamma^{-}$(s) for $s<0$ and $|s|$ sufficiently small, then
$-m^{+}\otimes\delta_{\xi_{d}=-\sqrt{1-R(x^{\prime},0,\xi^{\prime})}}=0$. This
implies $m^{+}=0$ and $H_{p}m=0$. As $m_{|x_{d}<0}=0$ and $\gamma^{+}(s)$ is
in $x_{d}<0$ for $s<0$ and $|s|$ sufficiently small, this implies that $m=0$
in a neighborhood of $\gamma^{+}(0)$.

####  A.4.3. Propagation at gliding points

We recall that a point $(x_{0}^{\prime},\xi_{0}^{\prime})\in
T^{*}\partial\Omega$ is is gliding or in ${\mathcal{G}}_{g}$, if
$R(x_{0}^{\prime},0,\xi_{0}^{\prime})-1=0$ and
$\partial_{x_{d}}R(x_{0}^{\prime},0,\xi_{0}^{\prime})>0$. Let $\gamma$ be the
integral curve of $H_{p}$ starting from
$(x^{\prime}_{0},0,\xi_{0}^{\prime},0)$. Then $\gamma(s)$ into $\\{x_{d}<0\\}$
for $s\not=0$ and $|s|$ sufficiently small. In a neighborhood of
$(x_{0}^{\prime},\xi_{0}^{\prime})$ in $T^{*}\partial\Omega$ all the point are
either hyperbolic, or gliding. We assume that
$j^{-1}\big{(}\gamma_{g}(s_{0};x_{0}^{\prime},\xi_{0}^{\prime})\big{)}\cap\operatorname{supp}m=\emptyset$
for $s_{0}<0$ where $|s_{0}|$ is sufficiently small. Here
$\gamma_{g}(s;x_{0}^{\prime},\xi_{0}^{\prime})=\Gamma(s;x_{0}^{\prime},\xi_{0}^{\prime})$,
then all the point $\rho$ in a neighborhood of
$j^{-1}\big{(}\gamma_{g}(s_{0};x_{0}^{\prime},\xi_{0}^{\prime})\big{)}$ are
not in the support of $m$. By continuity of $\Gamma$ the curve
$\Gamma(s;\rho)$ hit the boundary at $\rho^{\prime}$ in a neighborhood of
$(x_{0}^{\prime},\xi_{0}^{\prime})$. If $\rho^{\prime}$ is an hyperbolic
point, by the previous result the point $j^{-1}\big{(}\Gamma(s;\rho)\big{)}$
are not in the support of $m$. If $\rho^{\prime}$ is a gliding point, all the
points $\Gamma(s;\rho)$ are strictly gliding. In particular this implies that
$m$ is supported on $x_{d}=0$, then $1_{x_{d}>0}m=0$ and $m_{0}=0$. We can
apply Lemma A.5 and $m^{\partial}$ satisfied $H^{\prime}_{R}m^{\partial}=0$.
Let $\gamma_{g}$ be the integral curve of $H^{\prime}_{R}$ starting from
$(x_{0}^{\prime},\xi_{0}^{\prime})$. As by assumption
$m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}=m$ is 0 in a
neighborhood of
$j^{-1}\big{(}\gamma_{g}(s_{0};x_{0}^{\prime},\xi_{0}^{\prime})\big{)}$, we
have $m^{\partial}=0$ in a neighborhood of $\gamma_{g}(s_{0})$ and
$H^{\prime}_{R}m^{\partial}=0$, this implies that $\gamma_{g}(s)$ is not in
the support of $m^{\partial}$ in a neighborhood of $s=0$. As
$m=m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}$ we have $m=0$
in a neighborhood of $j^{-1}(x_{0}^{\prime},\xi_{0}^{\prime})$.

####  A.4.4. Propagation at diffractive points

We recall that a point $(x_{0}^{\prime},\xi_{0}^{\prime})\in
T^{*}\partial\Omega$ is in ${\mathcal{G}}_{d}$, if
$R(x_{0}^{\prime},0,\xi_{0}^{\prime})-1=0$ and
$\partial_{x_{d}}R(x_{0}^{\prime},0,\xi_{0}^{\prime})<0$. We keep the previous
notation for $\gamma$. For a point $\rho$ in a neighborhood of
$(x_{0}^{\prime},\xi_{0}^{\prime})\in T^{*}\partial\Omega\cup T^{*}\Omega$,
there are three cases, first the integral curve passing through $\rho$ hits
$x_{d}=0$ at an hyperbolic point and by previous result the integral curve is
not in support of $m$, second it does not hit $x_{d}=0$ and the integral curve
is not in the support of $m$ by propagation result in interior, third the
integral curve hits $x_{d}=0$ at a diffractive point. Then the support of
$m^{\partial}$ is in ${\mathcal{G}}_{d}$ and the support of $1_{x_{d}>0}m$ is
into $\\{(x,\xi),\ \xi_{d}^{2}+R(x,\xi^{\prime})=1,\text{ such that
}\gamma{(.;x,\xi)}\text{ hits }x_{d}=0,\ \ \xi_{d}=0\\}$.

We can apply Lemma A.4, then $m^{\partial}=0$ in a neighborhood of
$(x_{0}^{\prime},0,\xi_{0}^{\prime})$. As the integral curves hitting
$x_{d}=0$ at an hyperbolic points are not in the support of $1_{x_{d}>0}m$,
then $m_{0}$ is supported on $\xi_{d}=0$. We can apply Lemma A.5 to obtain
$m_{0}=0$. Then $H_{p}m=H_{p}(1_{x_{d}>0}m)=0$ and as, by assumption,
$\gamma(s)$ is not in the support of $m$ for $s<0$, $|s|$ sufficiently small,
we deduce that $\gamma(0)$ is not in the support of $m$.

####  A.4.5. Propagation at boundary: integral curves with high contact order

We recall that if $(x^{\prime}_{0},\xi^{\prime}_{0})$ is such that
$R(x_{0}^{\prime},0,\xi^{\prime}_{0},0)=1$,
$\partial_{x_{d}}R(x_{0}^{\prime},0,\xi^{\prime}_{0},0)=0$ and if we denote
the integral curve of $H_{p}$ starting from
$(x_{0}^{\prime},0,\xi^{\prime}_{0},0)$, by
$\gamma(s)=(x^{\prime}(s),x_{d}(s),\xi^{\prime}(s),\xi_{d}(s))$ . From the
assumption made (see Definition A.1) there exist $k\in\mathbb{N}$, $k\geq 3$
and $\alpha\neq 0$ such that $x_{d}(s)=\alpha s^{k}+{\mathcal{O}}(s^{k+1})$.
We denote $\gamma_{g}(s)=(x^{\prime}_{g}(s),\xi^{\prime}_{g}(s))$ the integral
curve of $H^{\prime}_{R}$, starting from $(x^{\prime}_{0},\xi^{\prime}_{0})$.
For each $k$ we assume that we have already proved that the integral curves
hitting $x_{d}=0$ at a point in ${\mathcal{G}}^{j}$ for $j<k$ or
${\mathcal{H}}$ are not in the support of $m$.

##### Case $k$ even, $\alpha<0$

The integral curve of $H_{p}$ starting from a point belonging to $T^{*}\Omega$
in a neighborhood of $(x^{\prime}_{0},\xi^{\prime}_{0})$ in
$T^{*}\partial\Omega\cup T^{*}\Omega$ eventually hits $x_{d}=0$ at a point
$\rho^{\prime}$, in ${\mathcal{H}}$ or ${\mathcal{G}}^{j}$ for $j\leq k$ (see
Section A.1 for definition of ${\mathcal{G}}^{j}$). By assumptions and by
induction this integral curve is not in the support of $m$ except if
$\rho^{\prime}$ is in ${\mathcal{G}}^{k}$, but in this case this integral
curve is in $x_{d}\leq 0$. This implies that $1_{x_{d}>0}m=0$, then $m_{0}=0$.
By Lemma A.5, we have $H^{\prime}_{R}m^{\partial}=0$ and as, by assumption,
$\gamma_{g}(s)$ is not in the support of $m^{\partial}$ for $s<0$, $|s|$
sufficiently small, we deduce that $\gamma_{g}(0)$ is not in the support of
$m^{\partial}$.

##### Case $k$ odd, $\alpha<0$

By the same argument as in previous case, the integral curve of $H_{p}$
starting from a point belonging to $T^{*}\Omega$ in a neighborhood of
$(x^{\prime}_{0},\xi^{\prime}_{0})$ in $T^{*}\partial\Omega\cup T^{*}\Omega$
hits ${\mathcal{G}}^{k}$ or is not in the support of $m$. Denote by
$\rho^{\prime}$ the point of this integral curve hitting $x_{d}=0$. The
generalized bicharacteristic starting from $\rho^{\prime}$ is on $x_{d}=0$ for
$s>0$ and in $x_{d}>0$ for $s<0$, and for $s>0$ all the points on the integral
curve of $H^{\prime}_{R}$ are in ${\mathcal{G}}_{g}$, if $|s|$ is sufficiently
small. As, by assumption the generalized bicharacteristic is not in support of
$m$ in the past, this means that $1_{x_{d}>0}m=0$ then $m_{0}=0$. We can apply
Lemma A.5 then $H^{\prime}_{R}m^{\partial}=0$. But $\gamma_{g}(s)$ is not in
support of $m=m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}$
for $s<0$ and $|s|$ sufficiently small as $\gamma_{g}(s)\in{\mathcal{G}}_{d}$,
then $\gamma_{g}(0)$ is not in the support of $m^{\partial}$.

##### Case $k$ even, $\alpha>0$

By induction, only the generalized bicharacteristics with the same order of
contact $k$ and the same sign condition $\alpha>0$ can be in the support of
$m$. Applying Lemma A.5 we have
$-\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime})m^{\partial}\otimes\delta_{x_{d}=0}\otimes\delta_{\xi_{d}=0}^{\prime}=0$
as by induction, $m^{\partial}=0$ when
$\partial_{x_{d}}R(x^{\prime},0,\xi^{\prime})\neq 0$. We deduce $H_{p}m=0$.
Then the propagation the support of $m$ is invariant by the flow of $H_{p}$.

##### Case $k$ odd, $\alpha>0$

By induction, only the generalized bicharacteristics with the same order of
contact $k$ and the same sign condition $\alpha>0$ can be in the support of
$m$. We can apply Lemma A.5, and by the same argument used in the previous
case, we have $H_{p}m=0$ and as $\gamma(s)$ is in $x_{d}<0$ for $|s|<0$
sufficiently small, $\gamma(s)$ is not in the support of $m$ for $s<0$ and by
propagation $\gamma(0)$ is not in support of $m$.

###### Proof of Proposition 5.1, first case.

By assumption for a point $\rho\in T_{b}\Omega=T^{*}\partial\Omega\cup
T^{*}\Omega$ for some $s_{0}\geq 0$, $\pi(\Gamma(s_{0},\rho))\in\omega$ and
from Proposition 4.3 and assumption (GCC), $m$ is null in a neighborhood of
$j^{-1}(\gamma(s_{0},\rho))$. As $\operatorname{supp}m$ is a closed set, if
$\rho\in\operatorname{supp}m$ there exist $s_{1}\in[0,s_{0}]$ such that
$j^{-1}\Gamma(s_{1},\rho)\cap\operatorname{supp}m\neq\emptyset$ and
$j^{-1}\Gamma(s,\rho)\cap\operatorname{supp}m=\emptyset$ for
$s\in(s_{1},s_{0}].$ At $\Gamma(s_{1},\rho)$ we can apply the results obtained
in this section to prove that
$j^{-1}\Gamma(s_{1},\rho)\cap\operatorname{supp}m=\emptyset,$ and reach a
contradiction. ∎

##  Appendix B Estimate of boundary trace, case strictly diffractive

In this section we prove a technical result allowing to estimate trace at
boundary in a neighborhood of a diffractive point. This precise estimate
allows to prove propagation of support in a neighborhood of such points.

###  B.1. Review on OIF and Airy functions

We give in this section some notation and results introduced in [13] following
Tataru [41].

######  Lemma B.1.

Let
$(x_{0}^{\prime},\xi_{0}^{\prime})\in\mathbb{R}^{d-1}\times\mathbb{R}^{d-1}$
be such that $R(x_{0}^{\prime},0,\xi_{0}^{\prime})-1=0$. For all $x_{d}$ in a
neighborhood of 0, there exist a smooth symplectic transformation
$\kappa:U_{0}\to U_{1}$ where $U_{0}$ and $U_{1}$ are some open set
respectively of
$\mathbb{R}^{d-1}_{x^{\prime}}\times\mathbb{R}^{d-1}_{\xi^{\prime}}$ and of
$\mathbb{R}^{d-1}_{y^{\prime}}\times\mathbb{R}^{d-1}_{\eta^{\prime}}$,
satisfying $(x_{0}^{\prime},\xi_{0}^{\prime})\in U_{0}$, $(0,0)\in U_{1}$,
$\kappa(x_{0}^{\prime},\xi_{0}^{\prime})=(0,0)$, and
$\kappa^{*}(\eta_{1})=R(x,\xi^{\prime})-1$. Moreover $x_{d}$ acts as a
parameter and $\kappa$ is smooth with respect $x_{d}=y_{d}$.

This lemma is classical. We can find a proof in Hörmander [24, Theorem
21.1.6]. This means we can complete the coordinate $R-1$ in a symplectic
manner.

To avoid ambiguity even if $x_{d}=y_{d}$, we denote $x_{d}$ when we work in
$(x,\xi^{\prime})$ variables and $y_{d}$ otherwise.

We call a symbol of order 0 a symbol $a\in S(1,|dx|^{2}+|d\xi^{\prime}|^{2})$
or in $S(1,|dy|^{2}+|d\eta^{\prime}|^{2})$. In this section we only use
tangential symbol, but as in what follows we have to use different classes of
symbols, we prefer use everywhere the same kind of notation.

Let $\chi$ be a smooth function supported in $U_{0}$, $\chi\geq 0$ and
$\chi=1$ in a neighborhood of $(x^{\prime}_{0},0,\xi^{\prime}_{0})$.

######  Lemma B.2.

Associated with $\kappa$, there exists $F$, a semiclassical Fourier Integral
Operator satisfying the following properties,

  1. i)

$F$ is a unitary operator uniformly with respect $x_{d}$.

  2. ii)

For all $\tilde{a}\in\mathscr{C}_{0}^{\infty}(U_{1})$,
$F^{-1}\operatorname{op_{sc}}(\tilde{a})F=\operatorname{op_{sc}}(a)$, where
$a=\kappa^{*}\tilde{a}+hb$ where $b$ is a symbol of order 0. In particular we
have
$F^{-1}\operatorname{op_{sc}}(\eta_{1}\tilde{\chi}^{2}(y,\eta^{\prime}))F=\operatorname{op_{sc}}(\chi^{2}(R-1))+h\operatorname{op_{sc}}(b)$,
where $\tilde{\chi}\in\mathscr{C}_{0}^{\infty}(U_{1})$, $\tilde{\chi}\geq 0$
and $b$ a symbol of order 0.

  3. iii)

There exist $\theta$ a symbol of order $0$, $B$ a bounded operator on $L^{2}$
such that $\operatorname{op_{sc}}(\theta)^{*}=\operatorname{op_{sc}}(\theta)$,
$\kappa^{*}\tilde{\chi}=\chi$ and
$(\partial_{x_{d}}F)F^{-1}=ih^{-1}\operatorname{op_{sc}}(\theta)+hB$.

  4. iv)

If the operators $A$ and $\tilde{A}$ are such that $A=F^{-1}\tilde{A}F$ then

$\partial_{x_{d}}A=F^{-1}\big{(}\partial_{y_{d}}\tilde{A}+ih^{-1}[\tilde{A},\operatorname{op_{sc}}(\theta)]+h[\tilde{A},B]\big{)}F$

where $B$ is the operator defined previously.

  5. v)

In particular we have $\kappa^{*}\\{\eta_{1},\theta\\}=\partial_{x_{d}}R$ in a
neighborhood of $(x^{\prime}_{0},0,\xi_{0}^{\prime})$.

For a proof of this lemma we refer to [13, Lemma B.2].

From now we shall use two semiclassical quantifications of symbol, one with
parameter $h$ and the other with parameter $h^{1/3}$. To avoid ambiguity or
confusion between both, we do not use the notation $\operatorname{op_{sc}}$
but we use classical quantification. For instance, for $a$ a symbol of order 0
we have $\operatorname{op_{sc}}(a)=\operatorname{op}(a(x,h\xi^{\prime}))$ that
is we keep the $h$ or $h^{1/3}$ in the notation.

Let $g=|dy|^{2}+h^{2/3}\langle
h^{1/3}\eta_{1}\rangle^{-2}|d\eta^{\prime}|^{2}$, this metric gives symbol
classes essentially as semiclassical symbol classes with $h^{1/3}$ for
semiclassical parameter. We let to the reader to check that $g$ is slowly
varying and $\sigma$ temperate. The "$h$" defined by Hörmander associated with
$g$ is $h^{1/3}\langle h^{1/3}\eta_{1}\rangle^{-1}$. It is the quantity we
gain in the asymptotic expansion for the symbol calculus. In particular the
function $\langle h^{1/3}\eta_{1}\rangle^{\nu}$ is a $g$-continuous and
$\sigma,g$ temperate for every $\nu\in\mathbb{R}$. We refer to [24, Chapter
18, Sections 4 and 5] for definitions used freely here.

To ease notation we write $|u|$ instead of $|u|_{L^{2}}(x_{d})$ when there is
no ambiguity on the fact that the $L^{2}$ norm is taken on variables
$x^{\prime}$ or $y^{\prime}$ at point $x_{d}$ or $y_{d}$. By the same abuse of
notation we write the inner product $(.|.)$ instead of
$(.|.)_{L^{2}(\mathbb{R}^{d-1})}(x_{d})$.

We recall some properties of the Airy function which is denoted by
$\operatorname{Ai}$. It verifies the equation
$\operatorname{Ai}^{\prime\prime}(z)-z\operatorname{Ai}(z)=0$ for
$z\in\mathbb{C}$, $\operatorname{Ai}$ is real on the real axis and
$\overline{\operatorname{Ai}(z)}=\operatorname{Ai}(\bar{z})$. Let
$\omega=e^{2i{\varepsilon}\pi/3}$ for ${\varepsilon}=\pm 1$.

For $x\in\mathbb{R}$, let $\alpha(x)=-\omega\operatorname{Ai}^{\prime}(\omega
x)/\operatorname{Ai}(\omega x)\in\mathscr{C}^{\infty}(\mathbb{R})$. As the
zero of $\operatorname{Ai}$ are on the negative real axis, the function
$\alpha$ is well defined for $x\in\mathbb{R}$ and smooth. The function
$\alpha$ satisfies the following properties.

######  Lemma B.3.

We have

  1. i)

$\alpha(x)=-\sqrt{x}+\frac{1}{4x}+b_{1}(x)$ for $x>0$

  2. ii)

$\alpha(x)={\varepsilon}i\sqrt{-x}+\frac{1}{4x}+b_{2}(x)$ for $x<0$

  3. iii)

$\operatorname{Re}\alpha(x)<0$ for all $x\in\mathbb{R}$

  4. iv)

$\alpha$ satisfies the differential equation
$\alpha^{\prime}(x)=\alpha^{2}(x)-x$,

where $b_{j}\in S(\langle x\rangle^{-5/2},|dx|^{2})$ for $j=1,2$.

For a proof of this lemma we refer to [13, Lemma B.3].

Let $\tilde{r}_{d}$ be such that $\kappa^{*}\tilde{r}_{d}=-\partial_{x_{d}}R$.
We assume that locally $\partial_{x_{d}}R<0$, this implies that
$\tilde{r}_{d}>0$ in a neighborhood of $(0,0)$.

Let
$\tilde{a}(y,\eta^{\prime})=h^{1/3}\tilde{\chi}(y,h\eta^{\prime})\tilde{r}_{d}^{1/3}(y,h\eta^{\prime})\alpha(\zeta)$
where $\zeta=h^{1/3}\eta_{1}\tilde{r}_{d}^{-2/3}(y,h\eta^{\prime})$. We assume
that on the support of $\tilde{\chi}(y,h\eta^{\prime})$,
$\tilde{r}_{d}(y,h\eta^{\prime})>0$. In what follows we denote
$\rho=(x,h\xi^{\prime})$ and $\tilde{\rho}=(y,h\eta^{\prime})$. We define
$\tilde{A}=\operatorname{op}(\tilde{a}(y,\eta^{\prime}))$,
$\tilde{\Psi}=h^{-1/3}\operatorname{op}(\langle
h^{1/3}\eta_{1}\rangle^{-1/2})$, and let $A=F^{-1}\tilde{A}F$,
$\Psi=F^{-1}\tilde{\Psi}F$.

We have $\tilde{a}\in S(h^{1/3}\langle h^{1/3}\eta_{1}\rangle^{1/2},g)$ as
$h^{-1/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2}\in S(h^{-1/3}\langle
h^{1/3}\eta_{1}\rangle^{-1/2},g)$ and from Lemma B.3. Observe that $\tilde{a}$
is bounded as on the support of $\tilde{\chi}(y,h\eta^{\prime})$,
$|\eta_{1}|\leq h^{-1}$ and $\langle h^{1/3}\eta_{1}\rangle^{1/2}\leq
h^{-1/3}$. We deduce that $\tilde{A}$ is bounded on $L^{2}$.

We state the following lemma proved in [13], see Lemma C.1.

######  Lemma B.4.

Let $\tilde{\chi}_{3}\in\mathscr{C}_{0}^{\infty}(U_{1})$ be such that
$(1-\tilde{\chi}_{3})\tilde{\chi}_{1}=0$ where
$\kappa^{*}\tilde{\chi}_{1}=\chi_{1}$. We have

$\displaystyle|\tilde{\Psi}\operatorname{op}(\tilde{\chi}_{1}(\tilde{\rho}))w|\lesssim|h^{-1/3}\operatorname{op}(\tilde{\chi}_{3}(\tilde{\rho})\langle
h^{1/3}\eta_{1}\rangle^{-1/2})\operatorname{op}(\tilde{\chi}_{1}(\tilde{\rho}))w|+|w|,$

for all $w$ in $L^{2}$.

###  B.2. Estimates at the boundary, glancing points

Let $\chi_{0}$ be a smooth cutoff in a neighborhood of
$(x^{\prime}_{0},0,\xi^{\prime}_{0})$ such that $\chi_{0}$ is supported in the
set where $\chi=1$ and $\chi_{0}=1$ in a neighborhood of
$(x^{\prime}_{0},0,\xi^{\prime}_{0})$. We consider the solution
$(r_{h},u_{h}^{d})$ solution of (4.47). We set ${\rm
r}=\operatorname{op_{sc}}(\chi_{0})r_{h}$ and ${\rm
v}=\operatorname{op_{sc}}(\chi_{0})u_{h}^{d}$. We have

$\displaystyle\|\operatorname{op_{sc}}(\chi_{0})(\operatorname{op_{sc}}((1-\chi)^{2})(R-1))r\|\lesssim
h\|r_{h}\|,$
$\displaystyle\|[\operatorname{op_{sc}}(\chi_{0}),\operatorname{op_{sc}}(\chi^{2}(R-1))]r\|\lesssim
h\|r_{h}\|,$
$\displaystyle\|[\operatorname{op_{sc}}(\chi_{0}),hD_{x_{d}}](r_{h},u_{h}^{d})\|\lesssim
h(\|r_{h}\|+\|u_{h}^{d}\|).$

Applying $\operatorname{op_{sc}}(\chi_{0})$ on System (4.47), we obtain the
system

(2.92) $\displaystyle hD_{x_{d}}{\rm r}+{\rm v}=h{\rm f}$ $\displaystyle
hD_{x_{d}}{\rm
v}-\operatorname{op}(\chi^{2}(x,h\xi^{\prime})(R(x,h\xi)-1)){\rm r}=h{\rm k},$
$\displaystyle\text{where }\|{\rm f}\|+\|{\rm
k}\|\lesssim\|q_{h}\|+\|f\|+\|r_{h}\|+\|u_{h}^{d}\|$ $\displaystyle\text{and
}{\rm r}=\operatorname{op_{sc}}(\chi_{0})r_{h},\ {\rm
v}=\operatorname{op_{sc}}(\chi_{0})u_{h}^{d}.$

######  Proposition B.1.

Let ${\rm r}$ and ${\rm v}$ satisfying system (2.92). There exists $C_{0}>0$
such that

$|i{\rm v}+A{\rm r}|^{2}(0)+\|\Psi(i{\rm v}+A{\rm
r})\|_{L^{2}(x_{d}>0)}^{2}\\\ \leq
C_{0}\big{(}\|u_{h}^{d}\|_{L^{2}(x_{d}>0)}^{2}+\|r_{h}\|_{L^{2}(x_{d}>0)}^{2}+\|{\rm
k}\|_{L^{2}(x_{d}>0)}^{2}+\|{\rm f}\|_{L^{2}(x_{d}>0)}^{2}\big{)}.$

###### Proof.

We compute

$\displaystyle\frac{1}{2}\partial_{x_{d}}|i{\rm v}+A{\rm r}|^{2}$
$\displaystyle=\operatorname{Re}(i\partial_{x_{d}}{\rm
v}+(\partial_{x_{d}}A){\rm r}+A\partial_{x_{d}}{\rm r}|i{\rm v}+A{\rm r})$
$\displaystyle=\operatorname{Re}(-h^{-1}(\operatorname{op}(\chi^{2}(\rho)(R(\rho)-1)){\rm
r}+h{\rm k})+(\partial_{x_{d}}A){\rm r}+ih^{-1}A(-{\rm v}+h{\rm f})|i{\rm
v}+A{\rm r})$ $\displaystyle=I_{1}+I_{2}+I_{3}+I_{4},$

where

$\displaystyle I_{1}=\operatorname{Re}(-h^{-1}A(i{\rm v}+A{\rm r})|i{\rm
v}+A{\rm r})$ $\displaystyle
I_{2}=\operatorname{Re}(h^{-1}(-\operatorname{op}(\chi^{2}(\rho)(R(\rho)-1)){\rm
r}+A^{2}{\rm r})|i{\rm v}+A{\rm r})$ $\displaystyle
I_{3}=\operatorname{Re}((\partial_{x_{d}}A){\rm r}|i{\rm v}+A{\rm r})$
$\displaystyle I_{4}=\operatorname{Re}(-{\rm k}+iA{\rm f}|i{\rm v}+A{\rm r}).$

We set ${\rm w}=F(i{\rm v}+A{\rm r})$. As $\tilde{A}$ and then $A$ are bounded
on $L^{2}$ we have $|{\rm w}|\lesssim|{\rm v}|+|{\rm r}|$. We now estimate the
terms $I_{k}$.

$\displaystyle I_{1}$ $\displaystyle=\operatorname{Re}(-h^{-1}FAF^{-1}{\rm
w}|{\rm w})$ $\displaystyle=\operatorname{Re}(-h^{-1}\tilde{A}{\rm w}|{\rm
w}).$

From Lemma B.3 we have

$-h^{-1}\operatorname{Re}\tilde{a}(y,\eta)\geq\delta
h^{-2/3}\tilde{\chi}^{2}(\tilde{\rho})\langle h^{1/3}\eta_{1}\rangle^{-1},$

and $h^{-1}\operatorname{Re}\tilde{a}(y,\eta)\in S(h^{-2/3}\langle
h^{1/3}\eta_{1}\rangle^{1/2},g)$.

Then from Fefferman-Phong inequality (see [24, Theorem 18.6.8]) and as the
real part of symbol of
$\operatorname{op}(h^{-1/3}\tilde{\chi}(\tilde{\rho})\langle
h^{1/3}\eta_{1}\rangle^{-1/2})^{*}\operatorname{op}(h^{-1/3}\tilde{\chi}(\tilde{\rho})\langle
h^{1/3}\eta_{1}\rangle^{-1/2})$ is
$h^{-2/3}\tilde{\chi}^{2}(\tilde{\rho})\langle h^{1/3}\eta_{1}\rangle^{-1}$
modulo an operator bounded on $L^{2}$, we have

$I_{1}\geq\delta|h^{-1/3}\operatorname{op}(\tilde{\chi}(\tilde{\rho})\langle
h^{1/3}\eta_{1}\rangle^{-1/2}){\rm w}|^{2}-C|{\rm w}|^{2},$

for $C>0$. From definition of ${\rm w}$ we have ${\rm
w}=\operatorname{op}(\tilde{\chi}_{0}(\tilde{\rho}))(iFu_{h}^{d}+\tilde{A}Fr_{h})+[\tilde{A},\operatorname{op}(\tilde{\chi}_{0}(\tilde{\rho}))]Fr_{h}$.
We can apply Lemma B.4 with
$\operatorname{op}(\tilde{\chi}_{1}(\tilde{\rho}))w=\operatorname{op}(\tilde{\chi}_{0}(\tilde{\rho}))(iFu_{h}^{d}+\tilde{A}Fr_{h})$
and as $[\tilde{A},\operatorname{op}(\tilde{\chi}_{0}(\tilde{\rho}))]$ is an
operator with symbol in $S(h^{2/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2},g)$.
Then
$\tilde{\Psi}[\tilde{A},\operatorname{op}(\tilde{\chi}_{0}(\tilde{\rho}))]$ is
a bounded operator on $L^{2}$. We then obtain

$I_{1}\geq\delta|\tilde{\Psi}\operatorname{op}(\tilde{\chi}_{0}(\tilde{\rho}))(iFu_{h}^{d}+\tilde{A}Fr_{h})|^{2}-C\big{(}|u_{h}^{d}|^{2}+|r_{h}|^{2}\big{)}.$

Same argument allows us to replace
$\operatorname{op}(\tilde{\chi}_{0}(\tilde{\rho}))(iFu_{h}^{d}+\tilde{A}Fr_{h})$
by ${\rm w}$. Finally we have

(2.93) $I_{1}\geq\delta|\tilde{\Psi}{\rm
w}|^{2}-C\big{(}|u_{h}^{d}|^{2}+|r_{h}|^{2}\big{)}.$

We have

$\displaystyle I_{2}$
$\displaystyle=\operatorname{Re}\big{(}h^{-1}F(-\operatorname{op}(\chi^{2}(\rho)(R(\rho)-1))+A^{2}){\rm
r}|{\rm w}\big{)}$
$\displaystyle=\operatorname{Re}\big{(}h^{-1}(-\operatorname{op}(h\eta_{1}\tilde{\chi}^{2}(\tilde{\rho}))+\tilde{A}^{2})F{\rm
r}|{\rm w}\big{)}+\operatorname{Re}\big{(}F\operatorname{op}(b(\rho)){\rm
r}|{\rm w}\big{)},$

where $\operatorname{op}(b(\rho))$ is bounded on $L^{2}$ (see Lemma B.2).

The symbol of $\tilde{A}^{2}$ is $\tilde{a}^{2}\in S(h^{2/3}\langle
h^{1/3}\eta_{1}\rangle,g)$ modulo a term in $S(h,g)$. From definition of
$\tilde{a}$ and Lemma B.3 we have

$\displaystyle\tilde{a}^{2}$
$\displaystyle=h^{2/3}\tilde{\chi}^{2}(\tilde{\rho})\tilde{r}_{d}^{2/3}(\tilde{\rho})\alpha^{2}(\zeta)$
$\displaystyle=h^{2/3}\tilde{\chi}^{2}(\tilde{\rho})\tilde{r}_{d}^{2/3}(\tilde{\rho})\big{(}\zeta+\alpha^{\prime}(\zeta)\big{)}$
$\displaystyle=h\eta_{1}\tilde{\chi}^{2}(\tilde{\rho})+h^{2/3}\tilde{\chi}^{2}(\tilde{\rho})\tilde{r}_{d}^{2/3}(\tilde{\rho})\alpha^{\prime}(\zeta).$

We have

$h^{-1/3}\tilde{\chi}^{2}(\tilde{\rho})\tilde{r}_{d}^{2/3}(\tilde{\rho})\alpha^{\prime}(\zeta)\in
S(h^{-1/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2},g),$

we then obtain

(2.94) $|I_{2}|\lesssim|{\rm r}|\big{(}|\tilde{\Psi}{\rm w}|+|{\rm
w}|\big{)}.$

We have

$\displaystyle I_{3}$
$\displaystyle=-\operatorname{Re}\big{(}F(\partial_{x_{d}}A)F^{-1}F{\rm
r}|{\rm w}\big{)}$
$\displaystyle=-\operatorname{Re}\big{(}(\operatorname{op}(\partial_{y_{d}}\tilde{a}(\tilde{\rho}))+ih^{-1}[\operatorname{op}(\tilde{a}(\tilde{\rho})),\operatorname{op}(\theta(\tilde{\rho}))]F{\rm
r}|{\rm w}\big{)}$
$\displaystyle\quad-\operatorname{Re}\big{(}h[\operatorname{op}(\tilde{a}(\tilde{\rho})),B]F{\rm
r}|{\rm w}\big{)},$

from Lemma B.2.

Observe that $\tilde{a}(\tilde{\rho})\in S(1,g)$ as $h^{1/3}\langle
h^{1/3}\eta_{1}\rangle^{1/2}$ is bounded on support of
$\tilde{\chi}(\tilde{\rho})$. Then
$\operatorname{op}(\partial_{y_{d}}\tilde{a}(\tilde{\rho}))$ and
$h[\operatorname{op}(\tilde{a}(\tilde{\rho})),B]$ are bounded operators on
$L^{2}$.

From properties of $\alpha$ and symbolic calculus, the symbol of

$h^{-1}[\operatorname{op}(\tilde{a}(\tilde{\rho})),\operatorname{op}(\theta(\tilde{\rho}))]\text{
is in }S(h^{-1/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2},g).$

Then we obtain

(2.95) $|I_{3}|\lesssim|{\rm r}|\big{(}|\tilde{\Psi}{\rm w}|+|{\rm
w}|\big{)}.$

We have

(2.96) $|I_{4}|=|\operatorname{Re}(-{\rm k}+iA{\rm f}|i{\rm v}+A{\rm
r})|\lesssim(|{\rm k}|+|{\rm f}|)(|{\rm v}|+|{\rm r}|).$

From (2.93)—(2.96) we have

$\displaystyle\frac{1}{2}\partial_{x_{d}}|i{\rm v}+A{\rm
r}|^{2}\geq\delta|\tilde{\Psi}{\rm w}|^{2}-C\big{(}|{\rm k}|^{2}+|{\rm
f}|^{2}+|u_{h}^{d}|^{2}+|r_{h}|^{2}\big{)}.$

Integrating this inequality between 0 and $\sigma>0$ we have

$|i{\rm v}+A{\rm r}|^{2}(0)+\delta^{\prime}\int_{0}^{\sigma}|\tilde{\Psi}{\rm
w}|^{2}(x_{d})dx_{d}\\\
\lesssim\|u_{h}^{d}\|_{L^{2}(x_{d}>0)}^{2}+\|r_{h}\|_{L^{2}(x_{d}>0)}^{2}+\|{\rm
k}\|_{L^{2}(x_{d}>0)}^{2}+\|{\rm
f}\|_{L^{2}(x_{d}>0)}^{2}+|u_{h}^{d}|^{2}(\sigma)+|r_{h}|^{2}(\sigma).$

Integrating this inequality between two positive values of $\sigma$ and as
$|\tilde{\Psi}{\rm w}|(y_{d})=|\Psi(i{\rm v}+A{\rm r})|(x_{d})$ we obtain the
result. ∎

We recall the notation
$\tilde{a}(y,\eta^{\prime})=h^{1/3}\tilde{\chi}(\tilde{\rho})\tilde{r}_{d}^{1/3}(\tilde{\rho})\alpha(\zeta)$
where ${\tilde{\rho}}=(y,h\eta^{\prime})$,
$\zeta=h^{1/3}\eta_{1}\tilde{r}_{d}^{-2/3}(\tilde{\rho})$ and by assumption,
$\tilde{r}_{d}(\tilde{\rho})>0$ on the support of
$\tilde{\chi}(\tilde{\rho})$. We state the following lemma, see [13, Lemma
B.5] for a proof.

######  Lemma B.5.

There exists a function $\beta\in\mathscr{C}^{\infty}(\mathbb{R})$ satisfying
the following properties

  1. i)

$-\beta^{\prime}-\beta\operatorname{Re}\alpha\geq 0$

  2. ii)

$\beta\in S(\langle x\rangle^{-1/4},|dx|^{2})$

  3. iii)

$\beta\gtrsim\langle x\rangle^{-1/4}$.

Let
$\tilde{c}(y,\eta^{\prime})=h^{-1/6}\tilde{\chi}_{2}(\tilde{\rho})\beta(\zeta)$,
where $\tilde{\chi}_{2}$ is supported on $\\{\tilde{\chi}=1\\}$ and
$\tilde{\chi}_{2}=1$ on a neighborhood of $(0,0)$. We have $\tilde{a}\in
S(h^{1/3}\langle h^{1/3}\eta_{1}\rangle^{1/2},g)$ and $\tilde{c}\in
S(h^{-1/6}\langle h^{1/3}\eta_{1}\rangle^{-1/4},g)$.

We define $\tilde{C}=\operatorname{op}(\tilde{c})$ and $C=F^{-1}\tilde{C}F$.

######  Proposition B.2.

Let ${\rm r}$ and ${\rm v}$ satisfying system (2.92). There exists $C_{0}>0$
such that

$|C(i{\rm v}+A{\rm r})|^{2}(0)\leq C_{0}\big{(}\|{\rm
k}\|_{L^{2}(x_{d}>0)}^{2}+\|{\rm
f}\|_{L^{2}(x_{d}>0)}^{2}+\|r_{h}\|_{L^{2}(x_{d}>0)}^{2}+\|u_{h}^{d}\|_{L^{2}(x_{d}>0)}^{2}\big{)}.$

###### Proof.

(2.97) $\displaystyle\frac{1}{2}\partial_{x_{d}}|C(i{\rm v}+A{\rm r})|^{2}$
$\displaystyle=\operatorname{Re}(\partial_{x_{d}}\big{(}C(i{\rm v}+A{\rm
r})\big{)}|C(i{\rm v}+A{\rm r}))$
$\displaystyle=\operatorname{Re}((\partial_{x_{d}}C)(i{\rm v}+A{\rm
r})+iC\partial_{x_{d}}{\rm v}+C\partial_{x_{d}}(A){\rm
r}+CA\partial_{x_{d}}{\rm r}|C(i{\rm v}+A{\rm r}))$
$\displaystyle=J_{1}+J_{2}+J_{3}+J_{4}+J_{5}=K_{1}+K_{2}+K_{3}+K_{4}+K_{5},$

where

$\displaystyle J_{1}$
$\displaystyle=\operatorname{Re}(C^{*}(\partial_{x_{d}}C)(i{\rm v}+A{\rm
r})|i{\rm v}+A{\rm r})$ $\displaystyle J_{2}$
$\displaystyle=\operatorname{Re}(\partial_{x_{d}}(A){\rm r}|C^{*}C(i{\rm
v}+A{\rm r}))$ $\displaystyle J_{3}$ $\displaystyle=\operatorname{Re}(-{\rm
k}+iA{\rm f}|C^{*}C(i{\rm v}+A{\rm r}))$ $\displaystyle J_{4}$
$\displaystyle=\operatorname{Re}(-ih^{-1}A{\rm v}|C^{*}C(i{\rm v}+A{\rm r}))$
$\displaystyle J_{5}$
$\displaystyle=\operatorname{Re}(-h^{-1}\operatorname{op}(\chi^{2}(\rho)(R(\rho)-1)){\rm
r}|C^{*}C(i{\rm v}+A{\rm r})).$

Taking as in the proof of Proposition B.2, ${\rm w}=F(i{\rm v}+A{\rm r})$ and
from Lemma B.2, we write

$\displaystyle K_{1}$
$\displaystyle=\operatorname{Re}(\tilde{C}^{*}(\partial_{y_{d}}\tilde{C}){\rm
w}|{\rm w})$ $\displaystyle K_{2}$
$\displaystyle=\operatorname{Re}((\partial_{y_{d}}\tilde{A}+h[\tilde{A},B])F{\rm
r}|\tilde{C}^{*}\tilde{C}{\rm
w})+\operatorname{Re}(F\operatorname{op}(b(\rho)){\rm
r}|\tilde{C}^{*}\tilde{C}{\rm w})$ $\displaystyle K_{3}$
$\displaystyle=\operatorname{Re}(-F{\rm k}+iFA{\rm
f}|\tilde{C}^{*}\tilde{C}{\rm w})$ $\displaystyle K_{4}$
$\displaystyle=\operatorname{Re}(-h^{-1}\tilde{A}{\rm
w}|\tilde{C}^{*}\tilde{C}{\rm
w})+\operatorname{Re}(\tilde{C}^{*}(ih^{-1}[\tilde{C},\operatorname{op}(\theta({\tilde{\rho}}))]+h\tilde{C}^{*}[\tilde{C},B])F{\rm
w}|{\rm w})$ $\displaystyle K_{5}$
$\displaystyle=\operatorname{Re}(h^{-1}(\tilde{A}^{2}-\operatorname{op}(h\eta_{1}\tilde{\chi}^{2}(\tilde{\rho})))F{\rm
r}|\tilde{C}^{*}\tilde{C}{\rm
w})+\operatorname{Re}((ih^{-1}[\tilde{A},\operatorname{op}(\theta({\tilde{\rho}}))]F{\rm
r}|\tilde{C}^{*}\tilde{C}{\rm w}).$

Observe that the symbol of $\tilde{C}^{*}(\partial_{y_{d}}\tilde{C})$ is in
$S(h^{-1/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2},g),$ then

(2.98) $|K_{1}|\lesssim|\tilde{\Psi}{\rm w}||{\rm w}|.$

From symbolic calculus the symbol of
$(\partial_{y_{d}}\tilde{A})^{*}\tilde{C}^{*}\tilde{C}$ is in $S(1,g)$, thus
this operator is bounded on $L^{2}$. Clearly the terms $[\tilde{A},B]$ coming
from remainder term of $\partial_{x_{d}}A$ (see iv) Lemma B.2) and
$\operatorname{op}(b(\rho))$ coming from remainder term of
$F\operatorname{op}(\chi^{2}(\rho)(R(\rho)-1))F^{-1}$ (see ii) Lemma B.2) are
bounded on $L^{2}$. As $\tilde{C}^{*}\tilde{C}$ has a symbol in
$S(h^{-1/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2},g)$, we obtain

(2.99) $|K_{2}|\lesssim|{\rm r}||{\rm w}|+|{\rm r}||\tilde{\Psi}{\rm w}|.$

From same arguments we have

(2.100) $|K_{3}|\lesssim(|{\rm k}|+|{\rm f}|)|\tilde{\Psi}{\rm w}|.$

To estimate the last term of $K_{4}$ we write
$\tilde{C}^{*}[\tilde{C},B]=\tilde{C}^{*}\tilde{C}B-\tilde{C}^{*}B\tilde{C},$
the first term gives a term estimated by $|\tilde{\Psi}{\rm w}||{\rm w}|$ and
the second is estimated by $|\tilde{C}{\rm w}|^{2}=(\tilde{C}^{*}\tilde{C}{\rm
w}|{\rm w})\lesssim|\tilde{\Psi}{\rm w}||{\rm w}|.$

To estimate the other terms of $K_{4}$ observe that the symbol of
$-h^{-1}\tilde{C}^{*}\tilde{C}\tilde{A}+ih^{-1}\tilde{C}^{*}[\tilde{C},\operatorname{op}(\theta({\tilde{\rho}}))])$
is
$-h^{-1}\tilde{c}^{2}\tilde{a}+h^{-1}\tilde{c}\\{\tilde{c},\theta(\tilde{\rho})\\}$
modulo a symbol in $S(h^{-2/3}\langle h^{1/3}\eta_{1}\rangle^{-1},g)$ and this
term can be estimate by $|\tilde{\Psi}w_{h}|^{2}.$ We compute

$\displaystyle h^{1/6}\\{\tilde{c},\theta(\tilde{\rho})\\}$
$\displaystyle=\\{\tilde{\chi}_{2}(\tilde{\rho}),\theta(\tilde{\rho})\\}\beta(\zeta)+\\{\beta(\zeta),\theta(\tilde{\rho})\\}\chi_{2}(\tilde{\rho})$
$\displaystyle=\\{\tilde{\chi}_{2}(\tilde{\rho}),\theta(\tilde{\rho})\\}\beta(\zeta)+\\{\zeta,\theta(\tilde{\rho})\\}\chi_{2}(\tilde{\rho})\beta^{\prime}(\zeta).$

The term
$h^{-1}\tilde{c}h^{-1/6}\\{\tilde{\chi}_{2}(\tilde{\rho}),\theta(\tilde{\rho})\\}\beta(\zeta)\in
S(h^{-1/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2},g),$ and the term of $K_{4}$
coming from this term can be estimated by $|\tilde{\Psi}{\rm w}||{\rm w}|$.
For the other term we have

$\displaystyle h^{-1/3}\\{\zeta,\theta(\tilde{\rho})\\}$
$\displaystyle=\\{\tilde{r}_{d}^{-2/3}(\tilde{\rho}),\theta(\tilde{\rho})\\}\eta_{1}+\\{\eta_{1},\theta(\tilde{\rho})\\}\tilde{r}_{d}^{-2/3}(\tilde{\rho}).$

The term
$h^{-1}\tilde{c}h^{-1/6}\chi_{2}(\tilde{\rho})\beta^{\prime}(\zeta)h^{1/3}\\{\tilde{r}_{d}^{-2/3}(\tilde{\rho}),\theta(\tilde{\rho})\\}\eta_{1}\in
S(h^{-1/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2},g)$ and the term of $K_{4}$
coming from this term can be estimate by $|\tilde{\Psi}{\rm w}||{\rm w}|$.

Thus, modulo remainder terms, the symbol of
$-h^{-1}\tilde{C}^{*}\tilde{C}\tilde{A}+ih^{-1}\tilde{C}^{*}[\tilde{C},\operatorname{op}(\theta)])$
is given by

$\displaystyle L=$
$\displaystyle-h^{-1}\tilde{c}^{2}\tilde{a}+h^{-1}\tilde{c}h^{-1/6}\tilde{\chi}_{2}(\tilde{\rho})\beta^{\prime}(\zeta)h^{1/3}\\{\eta_{1},\theta(\tilde{\rho})\\}\tilde{r}_{d}^{-2/3}(\tilde{\rho})$
$\displaystyle\qquad=-h^{-1}h^{-1/3}\tilde{\chi}_{2}^{2}(\tilde{\rho})\beta^{2}(\zeta)h^{1/3}\tilde{\chi}(\tilde{\rho})\tilde{r}_{d}^{1/3}(\tilde{\rho})\alpha(\zeta)$
$\displaystyle\qquad\quad-h^{-1}h^{-1/6}\tilde{\chi}_{2}(\tilde{\rho})\beta(\zeta)h^{-1/6}\tilde{\chi}_{2}(\tilde{\rho})\beta^{\prime}(\zeta)h^{1/3}\tilde{r}_{d}\tilde{r}_{d}^{-2/3}(\tilde{\rho})$

from v) of Lemma B.2. We thus obtain

$\displaystyle
L=h^{-1}\tilde{\chi}_{2}^{2}(\tilde{\rho})\beta(\zeta)\tilde{r}_{d}^{1/3}(\tilde{\rho})\big{(}-\beta(\zeta)\tilde{\chi}(\tilde{\rho})\alpha(\zeta)-\beta^{\prime}(\zeta)\big{)}\in
S(h^{-1},g).$

As $\tilde{\chi}$ is equal $1$ on the support of $\tilde{\chi}_{2}^{2}$,
$\beta\geq 0$ and
$-\beta(\zeta)\operatorname{Re}\alpha(\zeta)-\beta^{\prime}(\zeta)\geq 0$ we
have $\operatorname{Re}L\geq 0$. We can apply sharp Gårding inequality (see
[24, Theorem 18.6.7]), we yield, taking account remainder terms

(2.101) $K_{4}\geq-C\big{(}|\tilde{\Psi}{\rm w}|^{2}+|\tilde{\Psi}{\rm
w}||{\rm w}|\big{)}.$

Now we shall estimate $K_{5}$. The symbol of
$h^{-1}\tilde{A}^{2}+ih^{-1}[\tilde{A},\operatorname{op}(\theta(\tilde{\rho}))]-h^{-1}\operatorname{op}(h\eta_{1}\tilde{\chi}^{2}(\tilde{\rho}))$
is
$h^{-1}\tilde{a}^{2}+h^{-1}\\{\tilde{a},\theta(\tilde{\rho})\\}-\eta_{1}\tilde{\chi}^{2}(\tilde{\rho})$
modulo a symbol in $S(1,g)$. We have

$\displaystyle\\{\tilde{a},\theta(\tilde{\rho})\\}$
$\displaystyle=h^{1/3}\alpha(\zeta)\\{(\tilde{\chi}\tilde{r}_{d}^{1/3})(\tilde{\rho}),\theta(\tilde{\rho})\\}+h^{2/3}(\tilde{\chi}\tilde{r}_{d}^{1/3})(\tilde{\rho})\alpha^{\prime}(\zeta)\eta_{1}\\{\tilde{r}_{d}^{-2/3}(\tilde{\rho}),\theta(\tilde{\rho})\\}$
$\displaystyle\quad+h^{2/3}(\tilde{\chi}\tilde{r}_{d}^{-1/3})(\tilde{\rho})\alpha^{\prime}(\zeta)\\{\eta_{1},\theta(\tilde{\rho})\\}.$

The first two terms give terms estimated by $h^{1/3}\langle
h^{1/3}\eta_{1}\rangle^{1/2}\lesssim 1$ as on the support of $\tilde{\chi}$ we
have $|\eta_{1}|\lesssim h^{-1}$. Then both terms give associated operators
bounded on $L^{2}$. Modulo a bounded operator on $L^{2}$ we have to consider
the symbol, taking account v) of Lemma B.2

$\displaystyle
h^{-1/3}\tilde{\chi}^{2}(\tilde{\rho})\tilde{r}_{d}^{2/3}(\tilde{\rho})\alpha^{2}(\zeta)-h^{-1/3}(\tilde{\chi}\tilde{r}_{d}^{2/3})(\tilde{\rho})\alpha^{\prime}(\zeta)-\eta_{1}\tilde{\chi}^{2}(\tilde{\rho})$
$\displaystyle=h^{-1/3}\tilde{\chi}^{2}(\tilde{\rho})\tilde{r}_{d}^{2/3}(\tilde{\rho})\big{(}\alpha^{2}(\zeta)-\alpha^{\prime}(\zeta)-h^{1/3}\tilde{r}_{d}^{-2/3}(\tilde{\rho})\eta_{1}\big{)}$
$\displaystyle\quad-h^{-1/3}(\tilde{\chi}\tilde{r}_{d}^{2/3})(\tilde{\rho})(1-\tilde{\chi}(\tilde{\rho}))\alpha^{\prime}(\zeta).$

The first term is null from differential equation satisfying by $\alpha$ and
the value of $\zeta$. We claim that

(2.102)
$|\operatorname{op}\big{(}h^{-1/3}(\tilde{\chi}\tilde{r}_{d}^{2/3})(\tilde{\rho})(1-\tilde{\chi}(\tilde{\rho}))\alpha^{\prime}(\zeta)\big{)}F{\rm
r}|\lesssim|r_{h}|.$

The proof of the claim is given below. With this claim and what we do above,
the operator
$h^{-1}\tilde{A}^{2}+ih^{-1}[\tilde{A},\operatorname{op}(\theta(\tilde{\rho}))]-h^{-1}\operatorname{op}(h\eta_{1}\tilde{\chi}^{2}(\tilde{\rho})))$
gives a term bounded by $|r_{h}|.$ As the symbol of $\tilde{C}^{*}\tilde{C}$
is in $S(h^{-1/3}\langle h^{1/3}\eta_{1}\rangle^{-1/2},g),$ we obtain that

(2.103) $|K_{5}|\lesssim|r_{h}||\tilde{\Psi}{\rm w}|.$

From (2.97)—(2.101) and (2.103) we obtain

$\frac{1}{2}\partial_{x_{d}}|C(i{\rm v}+A{\rm r})|^{2}\gtrsim-\big{(}|{\rm
k}|^{2}+|{\rm f}|^{2}+|\tilde{\Psi}{\rm
w}|^{2}+|u_{h}^{d}|^{2}+|r_{h}|^{2}\big{)}.$

Integrating this inequality between 0 and $\sigma>0$, we have, estimating the
term coming from $|\tilde{\Psi}{\rm w}|$ by Proposition B.1,

(2.104) $|C(i{\rm v}+A{\rm r})|^{2}(0)\lesssim\|{\rm k}\|^{2}+\|{\rm
f}\|^{2}+\|u_{h}^{d}\|^{2}+\|r_{h}\|^{2}+|C(i{\rm v}+A{\rm r})|^{2}(\sigma).$

As

$|C(i{\rm v}+A{\rm r})|^{2}(\sigma)=(\tilde{C}^{*}\tilde{C}{\rm w},{\rm w}),$

we have

$|C(i{\rm v}+A{\rm r})|^{2}(\sigma)\lesssim|\tilde{\Psi}{\rm
w}|^{2}(\sigma)+|u_{h}^{d}|^{2}(\sigma)+|r_{h}|^{2}(\sigma).$

Integrating estimate (2.104) between two positive values of $\sigma$ and
estimating as above the term $|\tilde{\Psi}{\rm w}|^{2}(\sigma)$, we obtain
the conclusion of Proposition B.2. ∎

###### Proof of Claim (2.102).

Let $\chi_{1}$ smooth cutoff such that $\chi_{1}$ is supported on $\chi=1$ and
$\chi_{0}$ supported on $\chi_{1}=1$. We have $|{\rm
r}-\operatorname{op}(\chi_{1}(\rho))\operatorname{op}(\chi_{0}(\rho))r_{h}|\lesssim
h^{N}|r_{h}|$ for all $N>0$.

As $\kappa^{*}\tilde{\chi}_{1}=\chi_{1}$, from Lemma B.2 we thus have

$F^{-1}\operatorname{op}(\tilde{\chi}_{1}(\tilde{\rho}))F=\operatorname{op}(\chi_{1}(x,h\xi^{\prime}))+hK,$

where $K$ is bounded on $L^{2}$. We then have $F{\rm
r}=\operatorname{op}(\tilde{\chi}_{1}(\tilde{\rho}))F{\rm
r}+hK^{\prime}r_{h},$ where $K^{\prime}$ is bounded on $L^{2}$. Then

$\displaystyle\operatorname{op}\big{(}h^{-1/3}(\tilde{\chi}\tilde{r}_{d}^{2/3})(\tilde{\rho})(1-\tilde{\chi}(\tilde{\rho}))\alpha^{\prime}(\zeta)\big{)}F{\rm
r}$
$\displaystyle\quad=\operatorname{op}\big{(}h^{-1/3}(\tilde{\chi}\tilde{r}_{d}^{2/3})(\tilde{\rho})(1-\tilde{\chi}(\tilde{\rho}))\alpha^{\prime}(\zeta)\big{)}\big{(}\operatorname{op}(\tilde{\chi}_{1}(\tilde{\rho}))F{\rm
r}+hK^{\prime}r_{h}\big{)}.$

The first term coming from $\operatorname{op}(\tilde{\chi}_{1}(\tilde{\rho}))$
gives an operator with null symbol. As

$h^{-1/3}(\tilde{\chi}\tilde{r}_{d}^{2/3})(\tilde{\rho})(1-\tilde{\chi}(\tilde{\rho}))\alpha^{\prime}(\zeta)\in
S(h^{-1/3},g),$

then the second term is also bounded by $|r_{h}|$. This proves the claim. ∎

In the following proposition we keep notation introduced above, precise
statements on $\tilde{\chi}_{2}$ are given above Proposition B.2.

######  Proposition B.3.

Let $(x_{0}^{\prime},0)\in\partial\Omega$. Let
$\chi_{4}\in\mathscr{C}_{0}^{\infty}(\mathbb{R})$ be supported on a
neighborhood of 0 and $\chi_{4}=1$ in a neighborhood of 0. Let
$(x_{0}^{\prime},0,\xi_{0}^{\prime})$ and $U_{0}$ be as in the statement of
Lemma B.1. Let
$\ell\in\mathscr{C}_{0}^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d-1})$
supported on $\\{\chi_{2}=1\\}$ for every $x_{d}$, where
$\kappa^{*}\tilde{\chi}_{2}=\chi_{2}$. We moreover assume
$\partial_{x_{d}}R(x,\xi^{\prime})<0$ on support of $\ell$. We have

$\lim_{{\varepsilon}\to 0}\lim_{h\to
0}\Big{|}\Big{(}\operatorname{op}\big{(}\ell(x^{\prime},0,h\xi^{\prime})\chi_{4}\big{(}(R(x^{\prime},0,h\xi^{\prime})-1)/{\varepsilon}\big{)}(R(x^{\prime},0,h\xi^{\prime})-1)\big{)}{\rm
r}_{|x_{d}=0}|{\rm r}_{|x_{d}=0}\Big{)}_{\partial}\Big{|}=0.$

###### Proof.

Let ${\rm y}=(F{\rm r})_{|x_{d}=0}$. From Proposition B.2 and as ${\rm
v}_{|x_{d}=0}=0$ we obtain $|\tilde{C}\tilde{A}{\rm y}|$ is bounded. By
symbolic calculus and the support properties of $\tilde{\chi}$ and
$\tilde{\chi}_{2}$

$\tilde{C}\tilde{A}=h^{1/6}\operatorname{op}\big{(}\tilde{\chi}_{2}(\tilde{\rho})\beta(\zeta)\tilde{r}_{d}^{1/3}(\tilde{\rho})\alpha(\zeta)\big{)},$

modulo an operator with symbol in $S(h^{1/2}\langle
h^{1/3}\eta_{1}\rangle^{-3/4},g)$. From Lemma 4.4 this remainder term goes to
0 as $h$.

Let $\tilde{\ell}$ be such that $\kappa^{*}\tilde{\ell}=\ell$ and
$\kappa^{*}\chi_{4}(\eta_{1}/{\varepsilon})=\chi_{4}((R(x^{\prime},0,\xi^{\prime})-1)/{\varepsilon})$.
In what follow, to be coherent with previous notation, we define
$\tilde{\chi}_{4}=\chi_{4}$ and we use the notation $\tilde{\chi}_{4}$ when
the function is defined in $(y,\eta)$ variables.

From Lemma B.2 we have

$F^{-1}\operatorname{op}(h\eta_{1}\tilde{\ell}(\tilde{\rho})\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon}))F\\\
=\operatorname{op}\big{(}\ell(x^{\prime},0,h\xi^{\prime})\chi_{4}\big{(}(R(x^{\prime},0,h\xi^{\prime})-1)/{\varepsilon}\big{)}(R(x^{\prime},0,h\xi^{\prime})-1)\big{)}+h\operatorname{op}(r_{0}(x,h\xi^{\prime})),$

where $r_{0}$ is of order 0. The term coming from $r_{0}$ goes to 0 as $h$,
from Lemma 4.4. Then it is sufficient to prove that
$\displaystyle\lim_{{\varepsilon}\to 0}\displaystyle\lim_{h\to
0}\Big{(}\operatorname{op}(h\eta_{1}\tilde{\ell}(\tilde{\rho})\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})){\rm
y}|{\rm y}\Big{)}_{\partial}=0.$ Considering the symbol
$h\eta_{1}\tilde{\ell}(\tilde{\rho})\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})\in
S(h^{2/3}\langle h^{1/3}\eta_{1}\rangle,g)$ and from support properties of
$\tilde{\ell}$ and $\tilde{\chi}_{2}$, we have

$\displaystyle\operatorname{op}(h\eta_{1}\tilde{\ell}(\tilde{\rho})\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon}))=\tilde{\gamma}^{*}\operatorname{op}\big{(}h^{2/3}\eta_{1}\tilde{\ell}(\tilde{\rho})\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})\beta^{-2}(\zeta)\tilde{r}_{d}^{-2/3}(\tilde{\rho})|\alpha(\zeta)|^{-2}\big{)}\tilde{\gamma},$

where
$\tilde{\gamma}=\operatorname{op}\big{(}h^{1/6}\tilde{\chi}_{2}(\tilde{\rho})\beta(\zeta)\tilde{r}_{d}^{1/3}(\tilde{\rho})\alpha(\zeta)\big{)},$
modulo an operator with symbol in $S(h,g)$, then this last term involves a
term going to 0 as $h$ to 0. Then we obtain the following estimation, modulo a
term going to 0 as $h$ to 0,

(2.105)
$|\Big{(}\operatorname{op}(h\eta_{1}\tilde{\ell}(\tilde{\rho})\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})){\rm
y}|{\rm y}\Big{)}_{\partial}|\\\
\lesssim|\operatorname{op}\big{(}h^{2/3}\eta_{1}\tilde{\ell}(\tilde{\rho})\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})\beta^{-2}(\zeta)\tilde{r}_{d}^{-2/3}(\tilde{\rho})|\alpha(\zeta)|^{-2}\big{)}{\rm
z}\big{)}||{\rm z}|,$

where ${\rm
z}=\operatorname{op}\big{(}h^{1/6}\tilde{\chi}_{2}(\tilde{\rho})\beta(\zeta)\tilde{r}_{d}^{1/3}(\tilde{\rho})\alpha(\zeta)\big{)}{\rm
y}.$ Observe from Proposition B.2, $|{\rm z}|$ is bounded, as
$h^{1/6}\tilde{\chi}_{2}(\tilde{\rho})\beta(\zeta)\tilde{r}_{d}^{1/3}(\tilde{\rho})\alpha(\zeta)$
is the principal symbol of $\tilde{C}\tilde{A}$.

We claim that $h^{2/3}\eta_{1}\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})\in
S(\langle h^{1/3}\eta_{1}\rangle^{1/2},g).$ Indeed
$h^{2/3}|\eta_{1}|\lesssim\langle h\eta_{1}\rangle^{1/2}\langle
h^{1/3}\eta_{1}\rangle^{1/2},$ and this gives the sought estimate. For
derivatives and for $k\geq 1$, we have

$\partial_{\eta_{1}}^{k}\big{(}h^{2/3}\eta_{1}\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})\big{)}=kh^{2/3}(h/{\varepsilon})^{k-1}\tilde{\chi}_{4}^{(k-1)}(h\eta_{1}/{\varepsilon})+h^{2/3}\eta_{1}(h/{\varepsilon})^{k}\tilde{\chi}_{4}^{(k)}(h\eta_{1}/{\varepsilon}).$

As $h\eta_{1}$ is bounded on the support of
$\tilde{\chi}_{4}^{(k)}(h\eta_{1}/{\varepsilon})$ both terms are bounded by
$h^{k-1/3}$. From estimate $h^{2/3}\langle
h^{1/3}\eta_{1}\rangle\lesssim\langle h\eta_{1}\rangle$ we have

$h^{k-1/3}\lesssim h^{k/3}\big{(}\langle h\eta_{1}\rangle/\langle
h^{1/3}\eta_{1}\rangle\big{)}^{k-1/2},$

which proves the claim. Observe that the constants in above estimation are not
uniform with respect ${\varepsilon}$.

With the previous claim and as
$\beta^{-2}(\zeta)|\alpha(\zeta)|^{-2}\lesssim\langle
h^{1/3}\eta_{1}\rangle^{-1/2},$ we have

$L(y^{\prime},\eta^{\prime})=h^{2/3}\eta_{1}\tilde{\ell}(\tilde{\rho})\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})\beta^{-2}(\zeta)\tilde{r}_{d}^{-2/3}(\tilde{\rho})|\alpha(\zeta)|^{-2}\in
S(1,g).$

As $h^{2/3}|\eta_{1}|\lesssim|h\eta_{1}|^{1/2}\langle
h^{1/3}\eta_{1}\rangle^{1/2}\lesssim\sqrt{{\varepsilon}}\langle
h^{1/3}\eta_{1}\rangle^{1/2},$ on the support of
$\tilde{\chi}_{4}(h\eta_{1}/{\varepsilon})$, we deduce from Gårding inequality
(see [24, Theorem 18.6.7]) that the operator norm from $L^{2}$ to $L^{2}$ of
$\operatorname{op}(L)$ is bounded by
$C\sqrt{{\varepsilon}}+C_{\varepsilon}h^{1/3}$ where $C$ is independent of
${\varepsilon}$ and $C_{\varepsilon}$ may depend on ${\varepsilon}$. From that
and (2.105) we deduce the result. ∎

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# On Śankara Varman’s (correct) and Mādhava’s (incorrect) values for the
circumferences of circles

V. N. Krishnachandran

###### Abstract

This paper examines what computational procedures Śankara Varman (1774-1839)
and Sangamagrama Mādhava (c. 1340 - 1425), astronomer-mathematicians of the
Kerala school, might have used to arrive at their respective values for the
circumferences of certain special circles (a circle of diameter $10^{17}$ by
the former and a circle of diameter $9\times 10^{11}$ by the latter). It is
shown that if we choose the Mādhava-Gregory series for
$\frac{\pi}{6}=\arctan\frac{1}{\sqrt{3}}$ to compute $\pi$ and then use it
compute the circumference of a circle of diameter $10^{17}$ and perform the
computations by ignoring the fractional parts in the results of every
operation we get the value stated by Śankara Varman. It is also shown that,
except in an unlikely case, none of the series representations of $\pi$
attributed to Mādhava produce the value for the circumference attributed to
him. The question how Mādhava did arrive at his value still remains
unanswered.

2000 Mathematics Subject Classification : 01A32, 40–03, 40A25
Keywords and phrases : Kerala school of mathematics and astronomy, Śankara
Varman, Sadratnamāla, Sangamagrāma Mādhava, Mādhava-Gregory series,
computation of the value of $\pi$.

##  1 Introduction

Śankara Varman (1774-1839), an astronomer-mathematician of the Kerala school
of astronomy and mathematics, in his Sadrtnaml̄a has given a measure of the
circumference of a circle of diameter $10^{17}$. This value is correct to the
true value of the circumference rounded to the nearest integer. Earlier
mathematicians of the Kerala school like Nīlakaṇtha Somaȳjī and Śankara Vāriar
have quoted Sangamagrama Mādhava (c. 1340 - 1425), the founder of the Kerala
school, as stating that the circumference of a circle of diameter $9\times
10^{11}$ is $2827433388233$. But the true value of the circumference of this
circle when rounded to the nearest integer is $2827433388231$ and so Mādhava’s
value is in error by 2 units. This paper examines what computational
procedures these astronomer-mathematicians might have used to arrive at their
respective values for the circumferences of those special circles. It is shown
that if we choose the Mādhava-Gregory series for
$\frac{\pi}{6}=\arctan\frac{1}{\sqrt{3}}$ to compute $\pi$ and then use it
compute the circumference of a circle of diameter $10^{17}$ and perform the
computations by ignoring the fractional parts in the results of every
operation we get the value stated by Śankara Varman. It is also shown that,
except in an unlikely case, none of the series representations of $\pi$
attributed to Mādhava produce the value for the circumference attributed to
him. The question how Mādhava did arrive at his value still remains
unanswered.

In Section 2 we have a brief note on the life and work of Śankara Varman. In
the next three sections we discuss s̀ankara Varman’s value. We discuss it
first because the value is correct! In Section three we present the Sanskrit
verses with their English translations given in Sadratnamālā which give the
value of $\pi$ correct to 17 decimal places. Some issues like why a circle of
diameter $10^{17}$, what rounding method, etc. connected with the computation
of this value are discussed in Section 4. Sections 5 gives the complete
details of the possible methods of computation all of which can be carried out
by hand. In the last two sections we discuss the details of the computation of
Mādhava’s value by using the various infinite series representations of the
circumferences all attributed to Mādhava.

##  2 Śankara Varman

Śankara Varman (1774–1839), author of Sadratnamāla which is a treatise on
astronomy and mathematics composed in 1819, was an astronomer-mathematician
belonging to the Kerala school of astronomy and mathematics. He is considered
as the last notable figure in the long line of illustrious astronomers and
mathematicians in the Kerala school of astronomy and mathematics beginning
with Mādhava of Sangamagrāma (c. 1340 - c. 1425). Sadratnamala was composed in
the form of verses in Sanskrit in the traditional style followed by members of
the Kerala school even though India had been introduced to the western style
of writing books in mathematics at that time.

Śankara Varman was a close acquaintance of C. M. Whish (1794 – 1833) a civil
servant of East India Company attached to its Madras establishment. C. M.
Whish was the first person to bring to the attention of the European
mathematical scholarship the mathematical accomplishments of the Kerala school
of astronomy and mathematics (see [2]). Whish spoke of him and his work thus
(see [11]: “The author of Sadratnamalah is SANCARA VARMA, the younger brother
of the present Raja of Cadattanada near Tellicherry, a very intelligent man
and acute mathematician. This work, which is a complete system of Hindu
astronomy, is comprehended in two hundred and eleven verses of different
measures, and abounds with fluxional forms and series, to be found in no work
of foreign or other Indian countries.” For more details on the life and work
of Sankara Varman one may consult [8].

One of Śankara Varman’s contributions to mathematics was his computation of
the value of the mathematical constant $\pi$ correct to 17 decimal places.
This in itself, when compared to the computation of $\pi$ by William Shanks
correct to 507 decimal places in 1853, is of not much mathematical
significance. But, in the scenario and context of the nature of traditional
mathematical work in the early nineteenth century India in general and in
Kerala in particular, it acquires some significance and indicates that there
were persons at that time in Kerala who loved the labour of mathematical
computations for its own sake. This paper is an attempt to reproduce in its
totality the numerical computations that produced the value of $\pi$ correct
to 17 decimal places.

##  3 The Sanskrit verses in Sadratnamāla specifying the value of $\pi$

The verse 2 in Chapter 4 of Sadratnamala specifies the method of computation
of the circumference of a circle and also gives the circumference of a circle
having diameter $10^{17}$. The verse is reproduced below:

The verse has been translated into English as follows (see [7]):

> “Square the diameter of the circle, multiply by 12 and extract the square
> root. With this as the first term, develop the series thus: Divide this
> continuously by 3 and divide by 1, 3, 5, 7, 9, 11, … and form the terms. Add
> the odd terms and the even terms. Subtract the sum of the even terms from
> the sum of the odd terms. The result is the circumference of the circle.
>
> When this is done, the circumference of the big circle with diameter
> $10^{17}$ units is bhadrāmbudhisiddha-janma-gaṇita-śradhā sma yad bhūpagīḥ
> (by assignment of kaṭapaādi numerals).”

Let $D$ be the diameter of a circle and $C$ its circumference. Then what the
first half of the verse says can be put in the following form:

$\displaystyle C$
$\displaystyle=\left(\frac{\sqrt{12D^{2}}}{1}+\frac{\sqrt{12D^{2}}}{5\cdot
3^{2}}+\frac{\sqrt{12D^{2}}}{9\cdot 3^{4}}+\frac{\sqrt{12D^{2}}}{13\cdot
3^{6}}+\cdots\right)-$
$\displaystyle\phantom{=\,\,\,}\left(\frac{\sqrt{12D^{2}}}{3\cdot
3}+\frac{\sqrt{12D^{2}}}{7\cdot 3^{3}}+\frac{\sqrt{12D^{2}}}{11\cdot
3^{5}}+\frac{\sqrt{12D^{2}}}{15\cdot 3^{7}}+\cdots\right)$ (1)

This can be put succinctly in the following form:

$C=\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\sqrt{12D^{2}}}{(2k-1)3^{k-1}}$ (2)

The second half of the verse gives the value the circumference of a circle of
diameter $10^{17}$ (parārddha in Sanskrit). The value is presented in the
kaṭapayādi notation. When the Sanskrit letters in bhadrāmbudhisiddha-janma-
gaṇita-śradhā sma yad bhūpagīḥ are replaced by the corresponding numerals, we
get the following number:

$42\,39\,79\,85\,35\,62\,95\,14\,13$

As is well known, the Indian tradition is to write digits from right-to-left
which means that the digit in the unit’s place is written as the right most
digit and the digits with the highest place value is written as the left-most
digit. Writing the number in the current mathematical notation we get the
number as

$C^{\prime}=31\,41\,59\,26\,53\,58\,97\,93\,24.$

Sankara Varman claims that is the circumference of a circle whose diameter is
one parārdha which is the Sanskrit name for $10^{17}$.

The true value of $\pi$ correct to 20 decimal places is

$\pi=3.14159265358979323846.$

Hence, the circumference of a circle of diameter $10^{17}$ is

$\displaystyle C$ $\displaystyle=3.14159265358979323846\times 10^{17}$
$\displaystyle=31\,41\,59\,26\,53\,58\,97\,93\,23.846$ $\displaystyle\approx
31\,41\,59\,26\,53\,58\,97\,93\,24$ $\displaystyle=C^{\prime}$

Thus the value given by Sankara Varman agrees with the true value of the
circumference.

#### Remark 1

The result (1) is not due to S̄ankara Varman. The discovery of the result has
been attributed to Sangamagrāma Madhava the founder of the Kerala school of
astronomy and mathematics (see [1], [10] p.24).

#### Remark 2

Since $C=\pi D$, from Eq.(2), we have

$\frac{\pi}{6}=\sum_{k=1}^{\infty}(-1)^{k-1}\frac{1}{2k-1}\left(\frac{1}{\sqrt{3}}\right)^{2k-1}.$

which is the Mādhava-Gregory series for $\arctan x$ with
$x=\frac{1}{\sqrt{3}}$.

##  4 Some issues

In this context we would like to raise the following questions relating to
Sankara Varman’s statement in the cited verse.

###  4.1 Why a circle of diameter $10^{17}$?

The first question is of only historical curiosity: Why did Sankara Varman
consider a circle of diameter $10^{17}$? Is there any special significance to
the number $10^{17}$? There should definitely be some significance; he could
have easily chosen a circle of diameter, say, $10^{20}$. In several ancient
Indian astronomical texts one can see lists of names of powers of ten and
these lists stop with the number $10^{17}$ (see [5], [3] p.13). For example,
Srídhara (750) gives the following names: eka, daśa, em śata, sahasra, ayuta,
lakṣa, prayuta, koṭi, arbuda, abja, kharva, nikharva, mahâsaroja, śaṇku,
saritā-pati, antya, madhya, parârdha, and adds that the decuple names proceed
even beyond this.

Perhaps Sankara Varman chose the number $10^{17}$ because it is the largest
named number in Indian astronomical literature.

###  4.2 How did Sankara Varman verify that his value is the true value of the
circumference?

It seems that there is no indication in the available literature that suggests
that Sankara Varman did indeed verify that the value obtained by him is the
true value other than perhaps a repeat of the calculations. As we examine
below we have no details of how the computations were exactly carried out. Our
calculations as described in Section show that Sankara Varman obtained the
remarkable accuracy of 17 decimal places not by design but by chance.

###  4.3 How exactly were the computations performed?

The computation of the circumference of a circle of diameter $10^{17}$
involves the computation of $\sqrt{12\times(10^{17})^{2}}$. It may assumed
that Śankara Varman computed this value using Āryabhata’s method for the
computation of square roots. This method should have been completely familiar
to him because Āryabhatīya was known and had been thoroughly studied by Kerala
astronomers since at least the time of Sangamagrāma Mādhava (c. 1350 – c.
1425). One of Mādhava’s students, Vataśseri Parameśvaran Nambūdiri (c.
1380–1460), has written an extensive commentary on Āryabhatīya. Āryabhata’s
algorithm for the computation of square roots is also given Jyeṣṭhadeva’s
Yuktibhāṣa (see [9] p.161-162).

The following verse in Āryabhatīya describe the procedure for finding square
roots (see [6] p.36):

The verse has been translated as follows:

> “(Having subtracted the greatest possible square from the last odd place and
> then having written down the square root of the number subtracted in the
> line of the square root) always divide the even place (standing on the
> right) by twice the square root. Then, having subtracted the square (of the
> quotient) from the odd place (standing on the right), set down the quotient
> at the next place (i.e., on the right of the number already written in the
> line of the square root). This is the square root. (Repeat the process if
> there are still digits on the right.)”

An example illustrating the application of the above procedure is given in
Table 1. From the table we see that

$987654321=31426^{2}+60845.$

and so $\left\lfloor\sqrt{987654321}\right\rfloor=321426$.

Table 1: Details of the computation of $\sqrt{987654321}$ by Āryabhaṭa’s method Computations | Result | Notes
---|---|---
9,8 7,6 5,4 3,2 1 |  |
9 $-$ | 3 | $\lfloor\sqrt{9}\rfloor=\mathbf{3}$
9 |  | $\mathbf{3}^{2}=1$
0 8 $-$ | 3 1 | $\lfloor 8/(2\cdot 3)\rfloor=\mathbf{1}$
0 6 |  | $\mathbf{1}\cdot(2\cdot 3)$
2 7 $-$ |  |
2 6 1 |  | $\mathbf{1}^{2}=1$
2 6 6 $-$ | 3 1 4 | $\lfloor 266/(2\cdot 31)\rfloor=\mathbf{4}$
2 4 8 |  | $\mathbf{4}\cdot(2\cdot 31)=248$
0 0 1 8 5 $-$ |  |
0 0 0 1 6 |  | $\mathbf{4}^{2}=16$
0 0 1 6 9 4 $-$ | 3 1 4 2 | $\lfloor 1694/(2\cdot 314)\rfloor=\mathbf{2}$
0 0 1 2 5 6 |  | $\mathbf{2}\cdot(2\cdot 314)=1256$
0 0 0 4 3 8 3 $-$ |  |
0 0 0 0 0 0 4 |  | $\mathbf{2}^{2}=4$
0 0 0 4 3 7 9 2 $-$ | 3 1 4 2 6 | $\lfloor 43792/(2\cdot 3142)\rfloor=\mathbf{6}$
0 0 0 3 7 4 0 4 |  | $\mathbf{6}\cdot(2\cdot 3142)=37704$
0 0 0 0 0 6 0 8 8 1 |  |
0 0 0 0 0 0 0 0 3 6 |  | $\mathbf{6}^{2}=36$
0 0 0 0 0 6 0 8 4 5 |  | Remainder

###  4.4 Handling fractional parts

The computations involve, besides the operations of addition, subtraction and
multiplication, the operation of extraction of square root and division. The
square root to be extracted is $\sqrt{12\times(10^{17})^{2}}$. The number
$12\times(10^{17})^{2}$ is not an exact square and hence its square root is
not an exact integer. So some approximate value has to be used for further
numerical computations. The same is the case with divisions. The result of the
division of an integer by another integer is more often not an exact integer.
We can carry out the computations in several possible ways.

  1. 1.

Ignore the fractional part at every operation: We may perform the computation
by ignoring the fractions at every stage of the computation. This means that
that the result of an arithmetic operation involving positive integers shall
always be taken as the integral part of the result of the operation. Thus, in
this case we shall assume that the value of $\sqrt{12\times(10^{17})^{2}}$ is
$\lfloor\sqrt{12\times(10^{17})^{2}}\rfloor$ where $\lfloor\,\,\,\rfloor$
denotes the floor function.

Using the floor function, this computational scheme can be specified as
follows:

$\displaystyle x_{1}$ $\displaystyle=\left\lfloor\sqrt{12D^{2}}\right\rfloor$
$\displaystyle x_{k}$
$\displaystyle=\left\lfloor\frac{x_{k-1}}{3}\right\rfloor\text{ for
}k=2,3,4,\cdots$ $\displaystyle t_{k}$
$\displaystyle=\left\lfloor\frac{x_{k}}{2k-1}\right\rfloor\text{ for
}k=1,2,,3,4,\cdots$ $\displaystyle O$ $\displaystyle=t_{1}+t_{3}+t_{5}+\cdots$
$\displaystyle E$ $\displaystyle=t_{2}+t_{4}+t_{6}+\cdots$ $\displaystyle C$
$\displaystyle=O-E$

  2. 2.

Round the result of an operation to the nearest integer: Since, currently,
there are several different methods for rounding of numbers we may consider
one of the most commonly used methods of rounding which is rounding to the
nearest integer. If $x$ is a positive real number, the result of rounding $x$
to the nearest integer can be expressed as $\lfloor x+\frac{1}{2}\rfloor$. One
may round off the result of each division to nearest integer and take the
rounded value as the result of division.

Using the floor function, this computational scheme can be specified as
follows:

$\displaystyle x_{1}$
$\displaystyle=\left\lfloor\sqrt{12D^{2}}+\frac{1}{2}\right\rfloor$
$\displaystyle x_{k}$
$\displaystyle=\left\lfloor\frac{x_{k-1}}{3}+\frac{1}{2}\right\rfloor\text{
for }k=2,3,4,\cdots$ $\displaystyle t_{k}$
$\displaystyle=\left\lfloor\frac{x_{k}}{2k-1}+\frac{1}{2}\right\rfloor\text{
for }k=1,2,,3,4,\cdots$ $\displaystyle O$
$\displaystyle=t_{1}+t_{3}+t_{5}+\cdots$ $\displaystyle E$
$\displaystyle=t_{2}+t_{4}+t_{6}+\cdots$ $\displaystyle C$ $\displaystyle=O-E$

  3. 3.

Do the rounding at the end: In this method, one may retain the fractional
parts in all the terms and do the rounding at the last computation. However,
to start with, we have to assign a value for $\sqrt{12\times(10^{17})^{2}}$.

$\displaystyle x_{1}$ $\displaystyle=\sqrt{12D^{2}}$ $\displaystyle x_{k}$
$\displaystyle=\frac{x_{k-1}}{3}$ $\displaystyle\text{ for }k=2,3,4,\cdots$
$\displaystyle t_{k}$ $\displaystyle=\frac{x_{k}}{2k-1}$ $\displaystyle\text{
for }k=1,2,,3,4,\cdots$ $\displaystyle O$
$\displaystyle=t_{1}+t_{3}+t_{5}+\cdots$ $\displaystyle E$
$\displaystyle=t_{2}+t_{4}+t_{6}+\cdots$ $\displaystyle C$
$\displaystyle=\lfloor O-E\rfloor$ ignoring fractional part
$\displaystyle\text{or}\qquad C$ $\displaystyle=\left\lfloor
O-E+\tfrac{1}{2}\right\rfloor$ rounding to nearest integer

Even though this is a theoretical possibility, it is difficult to believe that
Śankara Varman did his computation using this method of rounding.

##  5 Computations

###  5.1 Ignore the fractional part at every operation

  1. Step 1.

Diameter of the circle is $D=10^{17}$.

  2. Step 2.

Square the diameter of the circle to get $D^{2}=(10^{17})^{2}=10^{34}$.

  3. Step 3.

Multiply by $12$ to get $12D^{2}=12\times 10^{34}$.

  4. Step 4.

Find the square root of $12D^{2}$. Applying Āryabhata’s algorithm and ignoring
the fractional part we get

$\sqrt{12D^{2}}=346410161513775458$

  5. Step 5.

With this as the first term, develop the series thus: Divide this continuously
by 3 and divide by 1, 3, 5, 7, 9, 11, … and form the terms. The results of
continuous division by $3$ are shown the second column of Table 1 and the
results of continuous division by 1, 3, 5, … are shown in the last column of
the same table. We have computed $38$ terms of the series in Eq.(2) because
the operations of successive divisions by $3$ produce the value $0$ for the
$38$-th term as can be seen from Table 1 and further divisions by $3$ will
only give 0’s.

  6. Step 6.

The following sums are calculated.

$\displaystyle\text{Sum of odd numbered terms }O$
$\displaystyle=346410161513775458+$
$\displaystyle\phantom{=0}1832857997427383+$
$\displaystyle\phantom{=0}129596020020118+\cdots$
$\displaystyle=354623317218212158$ $\displaystyle\text{Sum of even numbered
terms }E$ $\displaystyle=7698003589195010+$
$\displaystyle\phantom{=0}475185406740432+$
$\displaystyle\phantom{=0}36552723595417+\cdots$
$\displaystyle=40464051859232834$

  7. Step 7.

The circumference $C$ of a circle of diameter $10^{17}$ is given by

$\displaystyle C$ $\displaystyle=O-E$
$\displaystyle=354623317218212158-40464051859232834$
$\displaystyle=314159265358979324$

This is exactly the value given by Sankara Varman.

#### Remark

All the above computations can be done by hand and Śankara Varman would have
performed the computations definitely by hand and these computations would not
have been very cumbersome and tedious. However, for a modern scholar or a
student doing these computations would surely appear extremely complicated and
highly prone to error. Now they can verify the accuracy of the computations by
using any one of the several computer algebra systems available in the market
because such systems are good at performing exact arithmetic operation with
arbitrarily large integers. The author of the present paper did actually
verify the results using the Maxima (version 5.46.0) package which is a free
software released under the terms of the GNU General Public License. The
Maxima code for reproducing the results is given in Appendix 2.

Table 1: Computation of the terms in the series for the circumference
Term After dividing Division Sign After division position previous term by 3
by ${2k-1}$ by ${2k-1}$ ($k$) ($x_{k}$) ($t_{k}$) 1 346410161513775458 $(\div
1)\rightarrow$ $+$ 346410161513775458 2 115470053837925152 $(\div
3)\rightarrow$ $-$ 38490017945975050 3 38490017945975050 $(\div 5)\rightarrow$
$+$ 7698003589195010 4 12830005981991683 $(\div 7)\rightarrow$ $-$
1832857997427383 5 4276668660663894 $(\div 9)\rightarrow$ $+$ 475185406740432
6 1425556220221298 $(\div 11)\rightarrow$ $-$ 129596020020118 7
475185406740432 $(\div 13)\rightarrow$ $+$ 36552723595417 8 158395135580144
$(\div 15)\rightarrow$ $-$ 10559675705342 9 52798378526714 $(\div
17)\rightarrow$ $+$ 3105786972159 10 17599459508904 $(\div 19)\rightarrow$ $-$
926287342573 11 5866486502968 $(\div 21)\rightarrow$ $+$ 279356500141 12
1955495500989 $(\div 23)\rightarrow$ $-$ 85021543521 13 651831833663 $(\div
25)\rightarrow$ $+$ 26073273346 14 217277277887 $(\div 27)\rightarrow$ $-$
8047306588 15 72425759295 $(\div 29)\rightarrow$ $+$ 2497439975 16 24141919765
$(\div 31)\rightarrow$ $-$ 778771605 17 8047306588 $(\div 33)\rightarrow$ $+$
243857775 18 2682435529 $(\div 35)\rightarrow$ $-$ 76641015 19 894145176
$(\div 37)\rightarrow$ $+$ 24166085 20 298048392 $(\div 39)\rightarrow$ $-$
7642266 21 99349464 $(\div 41)\rightarrow$ $+$ 2423157 22 33116488 $(\div
43)\rightarrow$ $-$ 770150 23 11038829 $(\div 45)\rightarrow$ $+$ 245307 24
3679609 $(\div 47)\rightarrow$ $-$ 78289 25 1226536 $(\div 49)\rightarrow$ $+$
25031 26 408845 $(\div 51)\rightarrow$ $-$ 8016 27 136281 $(\div
53)\rightarrow$ $+$ 2571 28 45427 $(\div 55)\rightarrow$ $-$ 825 29 15142
$(\div 57)\rightarrow$ $+$ 265 30 5047 $(\div 59)\rightarrow$ $-$ 85 31 1682
$(\div 61)\rightarrow$ $+$ 27 32 560 $(\div 63)\rightarrow$ $-$ 8 33 186
$(\div 65)\rightarrow$ $+$ 2 34 62 $(\div 67)\rightarrow$ $-$ 0 35 20 $(\div
69)\rightarrow$ $+$ 0 36 6 $(\div 71)\rightarrow$ $-$ 0 37 2 $(\div
73)\rightarrow$ $+$ 0 38 0 $(\div 75)\rightarrow$ $-$ 0

###  5.2 Rounding to the nearest integer at every stage of computation

In this method computation we have (this may be verified by making appropriate
changes in the Maxima code given in Appendix 2)

$\displaystyle\sqrt{12D^{2}}$ $\displaystyle=346410161513775459$
$\displaystyle O$ $\displaystyle=354623317218212169$ $\displaystyle E$
$\displaystyle=40464051859232844$ $\displaystyle C$ $\displaystyle=O-E$
$\displaystyle=314159265358979325$

It may be noted that this value is different from the value given by Sankara
Varman. It differs by 1 in the unit’s place.

###  5.3 Do the rounding at the end

In this case (again, this may be verified using Maxima software) we have

$\displaystyle C$ $\displaystyle=314159265358979323.84375$
$\displaystyle\approx 314159265358979323$ (ignoring fractional part)
$\displaystyle\approx 314159265358979324$ (rounding to nearest integer)

The above result was also obtained by considering the first $38$ terms of the
series Eq.(2).

##  6 Sangamagrāma Mādhava

Sangamagrāma Mādhava, referred to as Mādhavan Ilañnppaḷli Emprān in a certain
manuscript, was a member of the Eṃprāntiri community which is a community of
Tuḷu speaking Brāhmins settled in Kerala. His forefathers are believed to have
migrated from the Tulu land to settle in Kudallur village, which is situated
on the southern banks of the Nila river not far from Tirunnavaya, a generation
or two before his birth and lived in a house known as Ilañnppaḷli whose
present identity is unknown (see [4]). There are no definite evidences to
pinpoint the period during which Mādhava flourished. From circumstantial
evidences historians have assigned the date c. 1340 - 1425 to Mādhava.

##  7 Mādhava’s value of $\pi$

A verse ascribed to Mādhava and quoted in the Kriyakramakri commentary of
Līlāvati by Śankara specifies a numerical value for $\pi$. The numerical value
is encoded in the bhūtasaṃkhyā system. In this system the numerals are
expressed by names of things, beings or concepts, which, naturally or in
accordance with the Śāstras, connote numbers; for example, the number two may
be denoted by anything which suggests the number 2. (For a list of words
commonly used to denote numbers see [3] p.14-17.)

The verse and an English translation are reproduced below:

> “The teacher Mādhava also mentioned a value of the circumference closer [to
> the true value] than that:
>
> vibudha [gods = 33] netra [eyes = 2] gaja [elephants = 8] ahi [serpents = 8]
> hutāśana [fires = 3] tri [three = 3] guna [qualities = 3] veda [Vedas = 4]
> bha [nakṣatras = 27] vārana [elephants = 8] bāhavāḥ [arms = 2 is the measure
> of a circle of diameter nava [9] nikharva [100,000,000,000].”

Thus, the circumference of a circle of diameter $9\times 10^{11}$ in the
Indian notational scheme is

$33\,\,2\,\,8\,\,8\,\,3\,\,\,3\,\,4\,\,27\,\,8\,\,2$

and in modern notational scheme is

$2,827,433,388,233.$

With this value for the circumference we have

$\displaystyle\pi$ $\displaystyle\approx\frac{2,827,433,388,233}{9\times
10^{11}}$
$\displaystyle\approx\underline{3.1415926535}\,\,\underline{92}22222\ldots$
(3)

and the true value of $\pi$ is

$\pi=3.1415926535\,\,89793238\ldots$

The approximate value of $\pi$ is correct up to the $10$-th decimal place.

With the true value of $\pi$, the circumference of a circle of diameter
$9\times 10^{11}$ is

$\displaystyle\pi\times 9\times 10^{11}$
$\displaystyle=2827433388230.81391461637904495\ldots$
$\displaystyle=2827433388231\qquad\text{rounding to nearest integer}$
$\displaystyle=2827433388230\qquad\text{ignoring the fractional part}$

With these approximate values for the circumference of a circle of diameter
$9\times 10^{11}$ we have

$\displaystyle\pi$ $\displaystyle\approx\frac{2827433388231}{9\times
10^{11}}=\underline{3.1415926535}\,\,\underline{90}\qquad\text{(this is
exact)}$ (4) $\displaystyle\pi$
$\displaystyle\approx\frac{2827433388230}{9\times
10^{11}}=\underline{3.1415926535}\,\,\underline{88}88888\ldots$ (5)

Both these values are closer to the true value of $\pi$ than the approximate
value given in Eq.(3).

##  8 How Mādhava might have computed the circumference?

Śankara Varman has indicated that he computed the value of $\pi$ using the
series in Eq.(1). But neither Nilakanṭha Somayāji nor Śankara Vāriar has
indicated how exactly Mādhava calculated the circumference of a circle of
diameter $9\times 10^{11}$. In this section we examine some of the possible
methods Mādhava might have employed to calculate the circumference.

The following formulae for computing the circumference of a circle of diameter
$D$ have been attributed to Mādhava.

  1. F1.

$\left(\dfrac{\sqrt{12D^{2}}}{1}+\dfrac{\sqrt{12D^{2}}}{5\cdot
3^{2}}+\dfrac{\sqrt{12D^{2}}}{9\cdot
3^{4}}+\cdots\right)-\left(\dfrac{\sqrt{12D^{2}}}{3\cdot
3}+\dfrac{\sqrt{12D^{2}}}{7\cdot 3^{3}}+\dfrac{\sqrt{12D^{2}}}{11\cdot
3^{5}}+\cdots\right)$ This is same as the series in Eq.(1) and it is the
series used by Śankara Varman to compute his value of $\pi$.

  2. F2.

$\dfrac{4D}{1}-\dfrac{4D}{3}+\cdots+(-1)^{n-1}\dfrac{4D}{2n-1}+(-1)^{n}4DF(n)$

Here $F(n)$ is a correction term used to give better approximation to the
circumference. Three expressions for $F(n)$ have been ascribed to Mādhava the
third being the one which gives the best approximation to the circumference.

    1. (a)

$F_{1}(n)=\dfrac{1}{4n}$

    2. (b)

$F_{2}(n)=\dfrac{n}{4n^{2}+1}$

    3. (c)

$F_{3}(n)=\dfrac{n^{2}+1}{n(4n^{2}+5)}$

  3. F3.

$3D+\dfrac{4D}{3^{3}-3}-\dfrac{4D}{5^{3}-5}+\dfrac{4D}{7^{3}-7}-\cdots$

  4. F4.

$\dfrac{16D}{1^{5}+4\cdot 1}-\dfrac{16D}{3^{5}+4\cdot
3}+\dfrac{16D}{5^{5}+4\cdot 5}-\dfrac{16D}{7^{5}+4\cdot 7}+\cdots$

###  8.1 Results of computations

####  8.1.1 Formula F1

Table 2: Circumference of a circle of diameter $9\times 10^{11}$ using formula
F1
Circumference ($C$) Number of Ignoring Operation-wise Final terms ($n$)
fractions rounding rounding 18 2827433388065 2827433388065 2827433388066 19
2827433388283 2827433388282 2827433388283 20 2827433388214 2827433388214
2827433388214 21 2827433388236 2827433388235 2827433388236 22 2827433388229
2827433388229 2827433388229 23 2827433388231 2827433388231 2827433388231 24
2827433388230 2827433388231 2827433388230 25 2827433388230 2827433388231
2827433388230 26 2827433388230 2827433388231 2827433388230 27 2827433388230
2827433388231 2827433388230

Under the specified conditions, the sum of the infinite series in F1 is the
underlined number in Table 2. Note that the series does not sum to Mādhava’s
value for the circumference; instead, it sums to a value much closer to the
true value of the circumference.

####  8.1.2 Formula F2

Table 3: Circumference of a circle of diameter $9\times 10^{11}$ using formula
F2 with correction term $F_{3}(n)$

| Circumference ($C$)
---|---
Number of | Ignoring | Operation-wise | Final
terms ($n$) | fractions | rounding | rounding
35 | 2827433388224 | 2827433388221 | 2827433388223
36 | 2827433388238 | 2827433388235 | 2827433388237
37 | 2827433388227 | 2827433388223 | 2827433388225
38 | 2827433388235 | 2827433388233 | 2827433388235
39 | 2827433388228 | 2827433388225 | 2827433388227
40 | 2827433388234 | 2827433388231 | 2827433388233
41 | 2827433388229 | 2827433388225 | 2827433388228
42 | 2827433388234 | 2827433388230 | 2827433388232
43 | 2827433388230 | 2827433388225 | 2827433388228
44 | 2827433388233 | 2827433388229 | 2827433388232
45 | 2827433388231 | 2827433388226 | 2827433388229
46 | 2827433388233 | 2827433388229 | 2827433388231
47 | 2827433388232 | 2827433388226 | 2827433388229
48 | 2827433388232 | 2827433388228 | 2827433388231
49 | 2827433388231 | 2827433388227 | 2827433388230
50 | 2827433388232 | 2827433388228 | 2827433388231
51 | 2827433388231 | 2827433388226 | 2827433388230
52 | 2827433388232 | 2827433388227 | 2827433388231
53 | 2827433388232 | 2827433388226 | 2827433388230
54 | 2827433388231 | 2827433388227 | 2827433388231
55 | 2827433388231 | 2827433388226 | 2827433388230
56 | 2827433388231 | 2827433388227 | 2827433388231
57 | 2827433388231 | 2827433388227 | 2827433388230
58 | 2827433388230 | 2827433388228 | 2827433388231
59 | 2827433388231 | 2827433388227 | 2827433388230
60 | 2827433388230 | 2827433388228 | 2827433388230
61 | 2827433388230 | 2827433388228 | 2827433388230
62 | 2827433388231 | 2827433388228 | 2827433388230
63 | 2827433388231 | 2827433388228 | 2827433388230
64 | 2827433388231 | 2827433388228 | 2827433388230
65 | 2827433388232 | 2827433388228 | 2827433388230

The numbers in boldface indicate the positions where Mādhava’s value for the
circumference appear. But, under any of the three methods of rounding, the
series does to sum to Mādhava’s value. Also, if the computations are done with
the rounding at the final stage then the sum converges to the value
$2827433388230$ (the underlined number in the table) which is different from
Mādhava’s value but closer to the true value than Mādhava’s value. The other
rounding procedures do not produce a repeating value even if we do
computations with number of terms up to a million. Moreover, if we take a
million terms, the term-by-rounding yields the sum $2827433387851$ which is a
very poor approximation to the true value of the circumference.

####  8.1.3 Formula F3

If we apply this formula, then the series indeed sum to a definite value.
Computations using Maxima software give the following results:

  * •

If we ignore all fractions then $C=2827433388211$ for all $n\geq 7663$. Thus
in this case the series sums to $2827433388211$. The number $7663$ is the
smallest integer $n$ for which

$\frac{4\times\times 9\times 10^{11}}{(2n-1)^{3}-(2n-1)}\geq 1.$

  * •

If we apply rounding at each operation then $C=2827433388236$ for $n\geq
9655$. SO, in this case the series sums to $2827433388236$. The number $9655$
is the smallest integer $n$ for which

$\frac{4\times\times 9\times 10^{11}}{(2n-1)^{3}-(2n-1)}+\frac{1}{2}\geq 1.$

  * •

If the rounding is done only at the last stage then $C=2827433388231$ for
$n\geq 8950$ and hence in this case the series sums to $2827433388231$.

Thus formula F3 also does not produce Mādhava’s value for the circumference.

####  8.1.4 Formula F4

Circumference of a circle of diameter $9\times 10^{11}$ using formula F4
Circumference ($C$) Number of Ignoring Operation-wise Final terms ($n$)
fractions rounding rounding 210 2827433388226 2827433388233 2827433388230 211
2827433388227 2827433388234 2827433388231 212 2827433388226 2827433388233
2827433388230 213 2827433388227 2827433388234 2827433388231 214 2827433388226
2827433388233 2827433388230 215 2827433388226 2827433388234 2827433388231 216
2827433388226 2827433388233 2827433388230 217 2827433388226 2827433388234
2827433388231 218 2827433388226 2827433388233 2827433388230 219 2827433388226
2827433388234 2827433388231 220 2827433388226 2827433388233 2827433388230 221
2827433388226 2827433388234 2827433388231 222 2827433388226 2827433388233
2827433388230 223 2827433388226 2827433388234 2827433388231 224 2827433388226
2827433388233 2827433388230 225 2827433388226 2827433388234 2827433388231 226
2827433388226 2827433388233 2827433388230 227 2827433388226 2827433388234
2827433388231 228 2827433388226 2827433388233 2827433388230 229 2827433388226
2827433388234 2827433388231 230 2827433388226 2827433388233 2827433388230 231
2827433388226 2827433388234 2827433388231 232 2827433388226 2827433388233
2827433388230 233 2827433388226 2827433388234 2827433388231 234 2827433388226
2827433388233 2827433388230 235 2827433388226 2827433388234 2827433388231 236
2827433388226 2827433388233 2827433388231 237 2827433388226 2827433388234
2827433388231 238 2827433388226 2827433388233 2827433388231 239 2827433388226
2827433388234 2827433388231 240 2827433388226 2827433388233 2827433388231 241
2827433388226 2827433388234 2827433388231 242 2827433388226 2827433388233
2827433388231 243 2827433388226 2827433388234 2827433388231 244 2827433388226
2827433388233 2827433388231 245 2827433388226 2827433388234 2827433388231 246
2827433388226 2827433388233 2827433388231 247 2827433388226 2827433388233
2827433388231 248 2827433388226 2827433388233 2827433388231 249 2827433388226
2827433388233 2827433388231 250 2827433388226 2827433388233 2827433388231

As in the case of formula F3, we note the following:

  * •

If we ignore the fractions the series in F4 sums to $2827433388226$ which is
different from Mādhava’s value.

  * •

If we perform rounding for each computation, the series in F4 sums to
$2827433388233$ which is precisely Mādhava’s value for the circumference.

  * •

If we perform rounding at the final stage only, the series in F4 sums to
$2827433388231$ which is different from Mādhava’s value but closer to the true
value of the conference.

###  8.2 Conclusion: Mādhava erred in his computations?

From the numerical computations described in Section 6.1 we arrive at the
following conclusions.

  * •

Among the four formulas considered only the following formula has produced
Mādhava’s value for the circumference:

$\dfrac{16D}{1^{5}+4\cdot 1}-\dfrac{16D}{3^{5}+4\cdot
3}+\dfrac{16D}{5^{5}+4\cdot 5}-\dfrac{16D}{7^{5}+4\cdot 7}+\cdots$

  * •

This series produces Mādhava’s value for the circumference only if the
computations are performed with rounding to the nearest integer at every stage
of the computation.

  * •

To ensure that the series indeed yields Mādhava’s value, the summation is to
be performed at leat up to the $247$-th term.

Rather than concluding that Mādhava arrived at his value of $2827433388233$
for the circumference as described above, a much simpler explanation might be
that Mādhava erred in his computation of the circumference of a circle of
radius $9\times 10^{11}$ however disquieting this explanation might be!

## References

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