# a visual introduction to the jacobi theta functions

Regrettably, as mathematics has slid toward the oubliette of abstractioneerism, special functions have kind of lost their glamour, and this is really rather sad because special functions are amazing and awesome.

So this is a first in a series of qualitative posts about special functions — rather than concentrating on either the quantitative features or the symmetries or special values, I’m going to try to give an intuitive picture — the sort that many high school mathematics curricula try very hard to give about the exponential and the trigonometrics — at least as much as the most that it is possible to get if one is paying attention and ignoring the idiocy of the calculus ceiling. (i.e. who decided long ago that the zenith of high school mathematics would be pebble betting…).

So, I’m going to assume that you’re familiar with the exponential function $\exp(z)=e^{z}$ and with the trigonometrics $\sin(z), \cos(z), \csc(z), \cot(z), \tan(z), \sec(z)$, and you know what the series representation of a function is. These functions have a real period — that is $\sin(z+2\pi)=\sin(z)$, and perhaps you have seen De Moivre’s theorem $e^{i \theta} = \cos(\theta) + i \sin(\theta)$.

The next sort of playing ground that (at least at the time I was in school) was the complex plane. It is rather sad that there’s no complex analysis taught in high schools — because it is a phenomenally rich and visual arena for play.

The roots of any jacobi theta function — when one keeps the nome fixed — are regularly spaced in a lattice whose aspect depends on the nome.

From mathworld “The Jacobi theta functions satisfy an almost bewilderingly large number of identities involving the four functions, their derivatives, multiples of their arguments, and sums of their arguments”, and also the quote “The theory of elliptic functions is the fairyland of mathematics. The mathe- mathematician who once gazes upon this enchanting and wondrous domain crowded with the most beautiful relations and concepts is forever captivated” is from.

So, herein, is a picture of a Jacobi theta function:

I’d like to give, in the next few posts, as best intuitive explanation as possible for why the texture of the geometry of Jacobi theta functions on the complex plane gives rise to such an abundance of identities.

# here’s a lark

So, consider the function

$G(q) = \prod_{n=1}^{\infty} \left[1 - \mathrm{agm}(q^{n^{2}}, q^{3^{n}}) \right]$

On the complex plane it looks like:

# fractional iteration, Ramanujan

Ramanujan stated (quaterly reports, Ramanujan’s Notebooks vol. 1) that if $F(x) = x^{2} -2$, $x>2$ then the fractional iterates of $F$ are:

$F^{1/2}(x) = \left( \frac{x + \sqrt{x^{2}-4}}{2}\right)^{\sqrt{2}} - \left( \frac{x - \sqrt{x^{2}-4}}{2}\right)^{\sqrt{2}}$

$F^{\log(3)/\log(2)}(x) = x^{3} - 3x$

$F^{\log(5)/\log(2)}(x) = x^{5} - 5x^{3} +5x$

what’s kind of amazing about this is that some of $F$‘s transcendental order compositions are polynomials with rational coefficients!

# functional inverse of the Y-combinator

The Y-combinator is a higher order function that computes the fixed point of functions.

$Y(d/dx) = Aexp(x)$

Problem: what is the inverse Y-combinator here, viz, suppose we make the $d/dx$ the unknown,

how could I solve:

given

$Y(\xi) = Aexp(x)$, solve for $\xi$