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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.

HyJu
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  • I don't follow you about the covering map $z \mapsto e^{2i \pi z}$. To me, this function is the modification of a weight-1/2 modular form $\theta(x) =\sum_{n=-\infty}^\infty e^{-\pi n^2x}$ periodic and (from the Poisson summation formula) having a functional equation $\theta(1/x)=x^{1/2}\theta(x)$. The same trick is applied to your function, and we get a more complicated group of transformations. So the idea behind this function is the Poisson summation formula and that $e^{-\pi t^2}$ is its own Fourier transform – reuns Feb 01 '17 at 13:39
  • @user1952009 I need more study!! Thanks. – HyJu Feb 01 '17 at 13:41
  • @user1952009 Thanks. I was aware of this particular PDE on real variables. Also about Paley-Wiener Theorem and similar ones, too. Actually I want to know whether $e^{\pi i n^2 \tau}e^{2\pi i z}$ constitute a set of solutions to a PDE on come compact domain in $\mathbb{C}^2$(we can find one such) and what the equation means. – HyJu Feb 01 '17 at 13:55
  • This is not an answer to your question, but my answer to this question here http://math.stackexchange.com/questions/814782/what-is-a-theta-function/814840#814840 may hold some interest for motivation of the origin of these functions. – Rene Schipperus Feb 01 '17 at 13:58
  • @ReneSchipperus Thanks for your comment. I'll try to understand the things you're saying in that document. – HyJu Feb 03 '17 at 17:42

2 Answers2

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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.

reuns
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  • Thanks for your answer. But I think it very slightly missed the point in my question. I need an understanding, not phenomenon. Phenomena in the nature are deceptive, as always. Thanks again! – HyJu Feb 03 '17 at 17:47
  • @HyJu A group acting on the space of analytic functions, the functions invariant under this action, Riemann surfaces, algebraic analysis, the eigenfunctions of the Fourier transform... Those are many things you'll want to look at from this function, that's what I wanted to show you – reuns Feb 03 '17 at 17:58
  • @HyJu This function is interesting because it introduces many new fruitful objects and concepts. That's what you need to understand – reuns Feb 03 '17 at 18:03
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$\theta (x, it)$ 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

Lada Dudnikova
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