Understanding Jacobi theta function I'm studying complex analysis and learned a bit about this interesting function:
$$ \Theta (z|\tau)=\sum_{n=-\infty}^{\infty}e^{\pi in^2\tau}e^{2\pi i n z}$$
for $\operatorname{Im}\tau>0$.
Before asking my question, let me introduce one of view points on this function first.
For any reasonable sequence $\left\{a_n\right\}_{n=-\infty}^{\infty}$, there is a function on the circle $F(e^{i\theta})=\sum_{n=-\infty}^{\infty}a_ne^{in\theta}$. We then ask whether this function can be holomorphically extended to an annulus around the unit circle in the plane $\mathbb{C}$. The result is the Laurent series $f(w)=\sum_{n=-\infty}^{\infty}a_nw^{n}$ whose restriction on $|w|=1$ is the original function $F$. This step is possible only if $a_n=O(r^{|n|})$ for some $0<r<1$, i.e., the sequences $a_n$ has at least an exponential decay as $|n| \to \infty$.
After obtained the function $f$, we then use the covering map $z \mapsto w=e^{2\pi i z}$ to full it back, making annulus into straight lines parallel to the x-axis.
The above $\Theta$ is obtained from the sequence $a_n=e^{\pi i n^2 \tau}$, which has very rapid decay(faster than any exponential) so that $\Theta$ is defined on the whole $z$-plane.
But you must agree that this approach is too formal and has no insight at all.
How would you explain this function? Is it a fundamental solution to a PDE in two complex variables? (If so, I also want to know where the differential equation comes from!)
Thanks.
 A: One of the motivation of this function is how it transforms under the Heisenberg group, making it closely related to the modular forms.


*

*$e^{-\pi x^2}$ is its own Fourier transform, thus for $b > 0$ and $ a \in \mathbb{R}$ we have the Fourier transform pair $$\varphi(x) = e^{-\pi b^2 x^2}e^{2i \pi a x}, \qquad \hat{\varphi}(\xi)= \frac{1}{b}e^{-\pi (\xi-a)^2/b^2}$$ 
by analytic continuation (being careful with the branched map $b^2 \mapsto b$) this stays true for $b \in \mathbb{C},Re(b^2) > 0$ 

*If everything converges (e.g. when $\varphi$ is a Schwartz function) we have the Poisson summation formula $$\sum_{n=-\infty}^\infty \varphi(n) =\sum_{n=-\infty}^\infty \hat{\varphi}(n)$$
Hence we get one of the Jacobi identities
$$\begin{eqnarray}\Theta\left(a | i b^2\right) &=& \sum_{n=-\infty}^\infty \varphi(n) 
= \sum_{n=-\infty}^\infty \hat{\varphi}(n)= \sum_{n=-\infty}^\infty \frac{1}{b}e^{-\pi (n-a)^2/b^2} \\ &=& \sum_{n=-\infty}^\infty \frac{e^{-\pi a^2/b^2}}{b}e^{-\pi n^2/b^2}e^{2\pi an/b^2} \\ &=&\frac{e^{-\pi a^2/b^2}}{b}\Theta\left(\frac{-ia}{b^2} | \frac{i}b^2\right)\end{eqnarray}$$
and by analytic continuation (being careful with the branched map $b^2 \mapsto b$) this functional equation stays true for every $a,b \in \mathbb{C}$ where $\Theta$ is analytic.
A: $\theta (x, it)$ is a solution of the Heat equation
                $\frac{\partial}{\partial t} \theta(x, it)= \frac{1}{4 \pi}\frac{ \partial^{2}}{\partial x ^{2}} \theta(x, it)$
Let's consider  $\theta(z,\tau)$
And substitute  $z=x \in \mathbb{R}, \tau=\{it : t \in \mathbb{R}, t > 0\}$
$\theta(x, it) = \sum_{n \in \mathbb{Z}} \exp(-\pi n^2 t) \exp(2\pi inx) = 1 + 2 \sum_{n \in \mathbb{Z}} \exp(-\pi n^2 t) \cos(2\pi nx)$
$\frac{ \partial}{\partial t} \theta(x, it) = 2   \sum_{n \in \mathbb{Z}} (- \pi n^2) \exp(-\pi n^{2} t) \cos(2 \pi n x)$
$\frac{ \partial^{2}}{\partial x ^{2}} \theta(x, it) = 2   \sum_{n \in \mathbb{Z}} (- 4 \pi^2 n^2) \exp(-\pi n^{2} t) \cos(2 \pi n x)$
However, I prefer to understand the theta functions as the one dimensional complex bundle implementation. See about it here
https://math.unice.fr/~beauvill/pubs/kyoto.pdf
