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On the Wolfram MathWorld page on Jacobi Theta Functions, it says that the Theta Functions are elliptic analogues of the exponential function. Is this because they satisfy certain properties that the exponential function satisfies? If so, which properties?

aleden
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  • periodicity${}$? – Angina Seng Jul 21 '18 at 16:18
  • I doubt its referring to periodicity since elliptic functions are all doubly periodic. It probably is a reference to elliptic functions being expressible in terms of theta functions, like sin and cos are in terms of exp. However, this is really a stretch since the relationship between them is not analogous to the relationship between e.g. sine and exp. – Grant B. Jul 21 '18 at 16:23
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    The analogy is that elliptic functions are the ratios of theta functions while circular functions are the ratios of exponential functions. But this is not a deep analogy. Theta functions are far more amazing than the exponential functions. – Paramanand Singh Jul 21 '18 at 16:23
  • See Paramanand's math notes for more info on these amazing functions :) – Grant B. Jul 21 '18 at 16:30
  • @GrantB.: Glad to know that someone liked those notes. – Paramanand Singh Jul 21 '18 at 16:43
  • Of course! You give an excellent introduction to elliptic and modular functions. They are what got me interested in this area. – Grant B. Jul 21 '18 at 16:59

2 Answers2

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The statement "Theta Functions are elliptic analogues of the exponential function" is not correct because it does not specify in which sense they are elliptic analogues. It is more accurate to state that the Jacobi elliptic functions (which are quotients of theta functions) are the elliptic analogues of trigonometric functions in the following sense. The Wikipedia article Jacobi elliptic functions states

The relation to trigonometric functions is contained in the notation, for example, by the matching notation sn for sin.

The Jacobi elliptic functions are doubly periodic which generalizes the single periodicity of the trigonometric functions which they reduce to. In fact, the current notation, along with the notation used by Jacobi is as follows: $$\textrm{sn}(u, k) := \sin( \textrm{am}(u, k)). $$ $$\textrm{cn}(u, k) := \cos( \textrm{am}(u, k)). $$ $$\textrm{dn}(u, k) := \Delta( \textrm{am}(u, k)). $$ When $\, k=0, \,$ then $\, \textrm{am}(u, 0) = u \,$ and this leads to $$ \textrm{sn}(u, 0) = \sin(u), \, \textrm{cn}(u, 0) = \cos(u), \, \textrm{dn}(u,0) = 1. $$ When $\, k=1, \,$ then $\,\textrm{am}(u,1) = \textrm{gd}(u),\,$ and this leads to $$ \textrm{sn}(u, 1) = \tanh(u), \, \textrm{cn}(u, 1) = \textrm{dn}(u, 1) = \textrm{sech}(u). $$ Thus, the Jacobi elliptic functions are a common generalization of the circular and hyperbolic trigonometric functions.

Somos
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This probably isn't a complete answer, because I'm no expert on $\vartheta$ functions, but from what I do know, they satisfy a couple of properties that are kind of reminiscent of the periodicity of exponential functions.

There seem to be a lot of $\vartheta$ functions on that page, but the definition I know is that $$\vartheta(z, \tau) = \sum_{n=- \infty}^{\infty}e^{\pi i n^2 \tau + 2 \pi i n z}$$

So these gadgets are defined in terms of ordinary exponential functions, and have some periodicity properties.

We have

$$\vartheta(z+1, \tau) = \vartheta(z, \tau)$$ since increasing $z$ by $1$ just spits out a $e^{2\pi i} = 1$ in every summand, and we also have for integer $\alpha, \beta$

$$ \vartheta(z + \alpha + \beta \tau, \tau ) = \vartheta(z, \tau) e^{- \pi i \beta^2 \tau - 2 \pi i \beta z} $$

which is a sort-of quasi-periodicity.

A. Thomas Yerger
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  • I see, so kind of how normal complex exponentials have one period corresponding to the unit circle, theta functions have 2 periods pertaining to a unit ellipse? – aleden Jul 21 '18 at 16:26
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    @aleden: theta functions are not doubly periodic, it is their ratios the elliptic functions which are doubly periodic. Also the elliptic functions are so called because they are obtained as inverse of certain integrals which are typically encountered in finding arc length of an ellipse. – Paramanand Singh Jul 21 '18 at 16:42
  • I see. I have seen lectures on how the jacobi elliptic functions are analogous to the trig functions though. – aleden Jul 21 '18 at 16:44
  • @aleden: elliptic functions are functions of two variables generally denoted by $u$ and $k$. When $k=0$ then these become same as trig functions and when $k=1$ they turn out to be hyperbolic functions. – Paramanand Singh Jul 21 '18 at 16:47