What is a Theta Function?

Question

What exactly is a theta function

$$\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z)= 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} \cos(2\pi n z) = \sum_{n=-\infty}^\infty q^{n^2}\eta^n"$$

and how does it arise naturally? (as in how does it just fall out of a computation illustrating the necessity for taking the trouble to give it a name?). Books like Whittaker-Watson simply say Elliptic functions can be represented as quotients of theta functions, them apparently being useful for explicit computations, so surely there is a natural way to see why this is so without knowing them in advance?

Note: Not as interested in the number theory motivation, more the elliptic integrals/functions and mathematical physics motivation,

Some sources say they are just Fourier series (of what?), other sources say they are quotients of Fourier series, other sources define them as an infinite product (which apparently hints at periodicity implying a useful Fourier representation), Goursat just call's it a Laurent series (of what?). Other sources say it is the Fourier series of a periodic version of some un-understandable function $\sigma$ appearing in elliptic functions (which doesn't seem to have anything to do with how one defines the theta function anyway).

Thus, assuming a theta function is a (quotient of?) Fourier series, what is it a Fourier series of, and how does it arise naturally?

Answer 1

They arise from an effort to find a product expansion of the elliptic functions by using the Weierstrass theorem to factor the zeros and poles. I hope the following outline gives an intuitive motivation for the creation of the theta functions. Naively we could write

$$\text{sn}(z)=z\pi \prod \frac{1-\frac{z}{2Kn+2K^{\prime}mi} }{ 1-\frac{z}{2Kn+2K^{\prime}mi +K^{\prime}i}}.$$

Define $\tau=\frac{K^{\prime}i}{K}$ then $$\text{sn}(2Kz)=2Kz\prod \left( \frac{1-\frac{z}{n+m\tau} }{1-\frac{z}{n+m\tau+\frac{1}{2}\tau}}\right).$$

However neither product $\prod\limits_{nm} 1-\frac{z}{n+m\tau}$ or $\prod\limits_{nm} 1-\frac{z}{n+m\tau+\frac{1}{2}\tau}$ converges absolutely. There are two ways to make convergent products out of these expresssions. One is to introduce convergence factors following Weierstrauss, and this leads to the $\sigma$ functions. Another way is to take the product first by $n$ and then by $m$. This results in the important theta functions. We define

$$\Theta_1(z)=2Kz \prod\limits_m \prod\limits_n \left( 1- \frac{z}{n+m\tau} \right).$$

After some manipulation and with $q=e^{\tau \pi i}$,

$$\Theta_1(z)=\frac{2K}{\pi}\sin(\pi z)\frac{\prod\limits_{m=1}^{\infty}1-2q^{2m}\cos(2z\pi)+q^{4m}}{\prod\limits_{m=1}^{\infty}\left[ 1-q^{2m} \right]^2}$$ is well defined with absolutely convergent numerator and denomenator. Similarly we define

$$\Theta_0(z)= \prod\limits_m \prod\limits_n \left( 1- \frac{z}{n+(m+\frac{1}{2})\tau} \right)$$ and obtain,

$$\Theta_0(z)=\frac{\prod\limits_{m=1}^{\infty}1-2q^{2m-1}\cos(2z\pi)+q^{4m-2}}{\prod\limits_{m=1}^{\infty}\left[ 1-q^{2m-1} \right]^2}.$$

As a result we have a product expression for the function $\text{sn}(2Kz)$.

$$\text{sn}(2Kz)=\frac{\Theta_1(z)}{\Theta_0(z)}.$$

Similarly we may define the functions,

$$\Theta_2(z)=\cos(\pi z)\frac{\prod\limits_{m=1}^{\infty}1+2q^{2m}\cos(2z\pi)+q^{4m}}{\prod\limits_{m=1}^{\infty}\left[ 1+q^{2m} \right]^2}$$ and $$\Theta_3(z)=\frac{\prod\limits_{m=1}^{\infty}1+2q^{2m-1}\cos(2z\pi)+q^{4m-2}}{\prod\limits_{m=1}^{\infty}\left[ 1+q^{2m-1} \right]^2}.$$

And we have

$$\text{cn}(2Kz)=\frac{\Theta_2(z)}{\Theta_0(z)}$$ $$\text{dn}(2Kz)=\frac{\Theta_3(z)}{\Theta_0(z)}.$$

Answer 2

Looking back, what I was really hoping for is the following simple explanation clearly delineating concepts:

A trigonometric (circular) function can be defined by focusing on the domain of the circle (thus there is a single period) or it's image. If we focus on the domain we analyze singly-periodic functions, defining trigonometric functions directly, while if we analyze the image we focus on arc length, defining trigonometric functions through the inverses of arc length integrals. Furthermore trigonometric functions can be defined in terms of exponential functions, as well as in terms of infinite products.

An elliptic function can similarly be defined in terms of the domain of an ellipse (thus there are two periods), giving doubly-periodic functions, or it's image, via inverse images of arc length integrals. If we do things directly this way we end up with Weierstrass elliptic functions, which have a double pole. However elliptic functions should be definable analogously to how trig functions are formulated in terms of exponentials, this is the origin of Jacobi Theta functions. My guess is that, just as trig functions are a sum of exponentials, since elliptic functions are some combination of theta functions on a parallelogram, this is why the elliptic functions have two distinct poles. Anyway, then a distinct formulation of elliptic functions in terms of infinite products gives us Weierstrass sigma functions.

I think that explains why every book is the way it is! Once you see these separate strands, only then should you go mixing them up, ahh that only took a few years to see...

Answer 3

Physically you can think of elliptic functions as describing the motion of a general frictionless pendulum. If the pendulum swings through an angle $\alpha$ then $k=\sin^2\tfrac12\alpha$ is the modulus, and $\alpha$ is called the modular angle. The motion of the pendulum is described by the function $\operatorname{sn}.$

Why does it have a real and imaginary period? Because if you reversed the direction of gravity, its motion would be that of a pendulum with the time variable $t$ replaced with $it$, roughly speaking, since the frequency of a pendulum depends on $\sqrt{g},$ where $g$ is the gravitational acceleration. Hence it must have two periods, one real, and one imaginary.

Theta functions are similarly thought of as the ratios of these elliptic functions or physically as solutions to the problem of heat conduction.

Answer 4

I'm not sure. But consider the heat equation on the circle. That is, we look for a solution $u$ to \begin{equation} \partial_t u(x,t) = \partial_x^2u(x,t). \end{equation} Here, $u$ is presumed to be a function on the circle times the positive time axis. Or rather, $u : \mathbb{R} \times \mathbb{R}_{> 0} \to \mathbb{R}$ is 1-periodic in the first variable. Assume that the solution is of the form $u(x,t) = A(x)B(t)$ where $A$ and $B$ are twice-differentiable and $A$ is $1$-periodic. Differentiating, we see that there is a constant $\lambda \in \mathbb{R}$ such that $A''(x) = \lambda A(x)$ and $B'(t) = \lambda B(t)$. Using the periodicity of $A$, we see that $\lambda \leq 0$. If $\lambda =0$, $A(x)$ must be constant (periodicity of $A$). If $\lambda < 0$, $A(x)$ must be of the form $a\cos(mx) + b \sin (mx)$ where $m^2= -\lambda$, and $m = 2\pi k$ for some $k \in \mathbb{Z}$ (periodicity of $A$). In particular, \begin{equation} B(t) = e^{-4\pi^2 k^2t} ,\ A(x) = \alpha e^{2i\pi kx} + \beta e^{-2i \pi kx} \text{ for some } \alpha, \beta \in \mathbb{C}. \end{equation} Thus, one is lead to the following solution of the heat equation on the circle: \begin{equation} u_k(x,t) = a_k e^{-4\pi^2 k^2t}e^{2i\pi k x} + a_{-k} e^{-4\pi k^2 t }e^{2i \pi kx}. \end{equation} Now, assuming we have complex numbers $(a_k)_{k \in \mathbb{Z}}$ which are the Fourier coefficients of some continuous, $1$-periodic function $f$ on $\mathbb{R}$, it is easy to show that the sum \begin{equation} u(x,t) = \sum_{k \in \mathbb{Z}} a_k e^{-4\pi^2k^2t}e^{2i\pi kx} \text{ where } t >0,\ x \in \mathbb{R} \end{equation} converges uniformly (for $t$ away from $0$) and satisfies the heat equation. Further, one can show that (Stein-Shakarchi, Fourier Analysis, Chapter 5, Corollary 3.4) \begin{equation} \lim_{t \to 0} u(x,t) = f(x) \text{ for all } x \in \mathbb{R}. \end{equation} Finally, it is easy to show that this solution of the heat equation is nothing but \begin{equation} f* H_t(x) := \int_0^1 f(y) H_t(x-y) dy \end{equation} where \begin{equation} H_t(x) := \sum_{k \in \mathbb{Z}} e^{-4\pi^2 k^2 t} e^{2i\pi k x} = \vartheta(x|4i\pi t). \end{equation} Thus, $\vartheta(x, 4i\pi t)$ is allegedly the Heat kernel for the circle. Another reference for this is Mumford's 'Lectures on Theta I' which describes the distributional convergence of $H_t(x)$ as $t \to 0$. From page 5 of this book I quote

Thus $\vartheta (x,it)$ may be seen as the fundamental solution to the heat equation when the space variable $x$ lies on a circle $\mathbb{R}/\mathbb{Z}$.