Questions about Weierstrass's elliptic functions

Question

From the wikipedia:

In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods $\omega_1$ and $\omega_2$ defined as

$$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_{n^2+m^2 \ne 0} \left\{ \frac{1}{(z+m\omega_1+n\omega_2)^2}- \frac{1}{\left(m\omega_1+n\omega_2\right)^2} \right\}. $$

Then $\Lambda=\{m\omega_1+n\omega_2:m,n\in\mathbb{Z}\}$ are the points of the period lattice, so that

$$ \wp(z;\Lambda)=\wp(z;\omega_1,\omega_2) $$

so I really like these elliptic functions. Much easier to understand than the jacobi ones, which I don't really get. Seems like such a nifty idea too. I don't get why these weren't the first ones that people thought of. Well, maybe they were but no one cared for them, because they didn't have applications like the jacobi ones. And it was only when people saw they had theoretical significance, that then they talked about them more. Or when they had to start teaching it, since these are easier to understand.

Well my first question is, why the exponent has to be $2$ and not $4$ or $6$ or any of the other even powers. Won't those powers generate convergent nonzero elliptic functions too? Then why do we only give attention to the ones, where the exponent is $2$? The wikipedia says any elliptic function can be written in terms of the corresponding Weierstrass one, that has the same periods. So maybe that's why. But I just find that shocking. So you're saying that we don't even need to give the terms in the Weierstrass series their own coefficient, that just polynomials (or other functions?) of the appropriate Weierstrass elliptical function will span the space of all elliptical functions with those periods? (even the ones that look like Weisterass series except that they have higher even-powered exponents?). I think someone has some explaining to do.

Secondly, why does no one mention the one dimensional analogues of Weierstrass elliptical functions. Those would be functions of one real variable, that are periodic. Why did we never study those kinds of periodic functions. Are they just not relevant to the theory of periodic functions of one real variable? Where unlike in the complex case, you can't generate all other periodic functions from those ones? I could see how you could get $\tan(x)$ out of those, maybe, since it is lucky enough to have poles at each period, but not the ones that don't have any poles (like $\sin$ and $\cos$).

Answer 1

It's right that the field of elliptic functions $K$ is generated by $\wp$ and$\wp^\prime$. To prove this, you consider the Laurent expansion of these two functions and use a comparison of coefficients and a variant of the Liouville theorem, which states that every elliptic function without a pole is constant. That wouldn't be possible if we used higher powers in the definition of $\wp$. If we restrict ourselves to the one-dimensional case, there isn't anything like the Liouville theorem, since there are everywhere differentiable, bounded functions, who are nonconstant.

Answer 2

Historically, after Legendre's work on Elliptic integrals; Gauss, Abel, Jacobi, Eisenstein, and others studied elliptic functions each in their own way. Weierstrass came much later in the 19th century with his own version. Ramanujan also had his own version of elliptic functions. What seems simple, natural and understandable to you is not the same for everyone.

To answer your first question, define the double sum $\ p_k(z) := \sum_{n,m\in\mathbb{Z}} (z+m\omega_1+n\omega_2)^{-k}.\,$ Then $\,p_2(z) \!=\! \wp(z),\,$ $\,p_3(z) \!=\!-\frac12 \wp'(z),\,$ $\,p_4(z) \!=\! \wp(z)^2\! -\!\frac1{12} g_2,\,$ $\,p_5(z) \!=\! -\frac12 \wp(z)\wp'(z),\,$ $\,p_6(z) \!=\! \wp(z)^3 \!-\!\frac3{20} g_2\wp(z) - \frac1{10}g_3,\,$ and so on with the caveat that $\,p_2(z)\,$ summation needs to be rewritten in order to be convergent. These follow from the equation $\ p_k\!'(z) = -k\ p_{k+1}(z).\ $ The answer to your question is that these sums are elliptic functions and can be expressed as polynomials in $\, p_2(z)\,$ and $\,p_3(z)\,$ (or equivalently $\,\wp(z)\,$ and $\,\wp'(z)$).

To answer your second question about one dimensional analogues. Yes, it's possible to develop that theory, but it's not that well known. Fix one real period and let the other non-real period approach infinity. You get simply periodic meromorphic functions like $\cot(\pi z).$ For some details read MSE question 1849878 "How did Euler prove the partial fraction expansion of the cotangent function?"