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proof of functional equation for the theta function

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\begin{document}
All sums are over all integers unless otherwise specified.  Thus the 
theta function is $$\theta(x)=\sum_ne^{-\pi n^2x}.$$

Using the 
\PMlinkname{Jacobi's identity for $\vartheta$ functions}{JacobisIdentityForVarthetaFunctions}  with $z=0$ and 
$\tau = i/x$,
so that $-1/\tau = ix$ gives
$$
\theta_3 ( 0 \mid ix ) = (1/x)^{1/2} \theta_3(0 \mid i/x).
$$

Using the \PMlinkname{definition of $\theta_3$}{JacobiVarthetaFunctions} we have that the left hand side is
$$
\sum_{n} e^{-\pi x n^2} = \theta(x)
$$
while the right hand side is
$$
(1/x)^{1/2}\sum_{n} e^{i\pi (i/x)n^2}
$$
which is

$$
(1/x)^{1/2}\sum_{n} e^{-\pi n^2/x} =  \frac{1}{\sqrt{x}}\theta (1/x)
$$
so the identity is established.

The identity is attributed to Poisson by Jacobi \cite{Jacobi}. Jacobi writes:
M. Poisson, dans ses savantes recheches sur les int\'egrales d\'efinies, a
fait conna\^itre plusieurs propri\'et\'es de la fonction $\Theta(x)$. Les m\'ethodes
d\'elicates, propres \`a cet illustre g\'eom\`etre, trouvent une belle v\'erification
dans la th\'eorie des fonctions elliptiques. Par exemple, M. Poisson d\'emontre
dans dix-neuvi\`eme cahier du Journal de l'\'ecole polytechnique la formule remarquable:
$$
\sqrt{\frac{1}{x}} = \frac{1+2e^{-\pi x} + 2e^{-4\pi x} + 2 e^{-9\pi x}  +2e^{-16\pi x}+ \cdots}{1+2e^{-\frac{\pi}{x}} + 2e^{-\frac{4\pi}{x}} + 2 e^{-\frac{9\pi}{x}} + 2e^{-\frac{16\pi}{x}}+\cdots}
$$

\begin{thebibliography}{99}
\bibitem{Jacobi} M.C.G.J Jacobi, \emph{Notices sur Les Fonctions Elliptiques},
in Jacobi's Gesammelte Werke, Band 1, Berlin, 1881, page 260.
\end{thebibliography}
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