# Questions tagged [elliptic-functions]

Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.

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### Connection between Lame equation with Weierstrass and elliptic sine

Lame function is the solution of the following equation, $$\frac{d^2w}{dz^2}+\left(A+B\wp(z)\right)w=0,$$ where $A$ and $B$ are constants and $\wp(z)$ is the Weierstrass elliptic function. Wiki says ...
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### How to prove the infinite product expression of this Jacobian elliptic $\operatorname{sn}$ function?

How to prove the following infinite product expression? \begin{align*} \operatorname{sn}(x,k)=\tanh \left(\frac{\pi\,\!x}{2 K(k')}\right)\prod\limits_{n=1}^{+\infty\,}\frac{ \tanh \left(\frac{\pi\left(...
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### Weierstrass factorization of $\operatorname{sn}$ and others

By the Weierstrass factorization theorem, we can write the elliptic function $\operatorname{sn}$ as $$\operatorname{sn}u=\frac{B}{A}$$ where $B$ is a certain product over its zeros and $A$ is a ...
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Jacobi defined the elliptic function by $u=\int_0^{\phi}\frac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}$ ,$am(u)=\phi,sn(u)=\sin(am(u)),cn(u)=\cos(am(u))$ $dn(u)=\sqrt{1-k^2sn^2(u)},k'^2+k^2=1$ and he set $... 1 vote 0 answers 35 views ### Given two points$P$and$Q$on a fundamental parallelogram, construct an elliptic function with simple poles at$P, Q$by contour integral of$\wp$We know that every elliptic functions can be written as a rational function of$\wp$and$\wp'$, where$\wp$is the Weierstrass p function.. However, I am wondering how, given two arbitrary points$P$... 3 votes 1 answer 86 views ### If two elliptic functions share the same poles and zeros (including multiciplity) then they are proportional I’m trying to understand the following statement found on my lecture notes: If two elliptic functions share the same poles and zeros (including multiciplity) then they are proportional I’m trying to ... 11 votes 2 answers 291 views ### Integrals of$\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k$Consider a type of integrals $$\int_{0}^{1} \left ( \frac{K^\prime}{K} \right )^{s-1} f(k)\text{d}k$$ where$K=K(k),K^\prime=K(\sqrt{1-k^2})$are complete elliptic integrals, and$k$is an elliptic ... 2 votes 1 answer 40 views ### Using the first Liouville theorem The first Liouville theorem states Any elliptic function without poles is constant. Let$\wp$be the Weierstrass elliptic function of a lattice$L$and let$\sigma$be the Weierstrass sigma function ... 1 vote 0 answers 54 views ### Show "hexagonal elliptic curves" have algebraic coefficients Introduction: Given any$\tau \in \mathbb H$, there is an elliptic curve$y^2=4x^3-g_2(\tau)x-g_3(\tau)$which has$\{1,\tau\}$as its lattice. Wikipedia gives an "explicit" relation between ... 2 votes 1 answer 78 views ### How to manipulate differential equation into Weierstrass P form? I have the diffeq$y'^2=\frac23 y^3+\alpha$for some$\alpha\in\mathbb R$. This has two solutions in the form of a Weierstrass P elliptic function which are $$6^{1/3}\wp\left(\frac{x\pm C}{6^{1/3}};0, ... 1 vote 0 answers 68 views ### Division of Bernoulli's lemniscate by ruler and compass Bernoulli's lemniscate with unit half-width is given by the polar equation r=\sqrt{\cos 2\theta}. Using this curve, in the first quadrant, we define the function \operatorname{arcsl} according to ... 4 votes 1 answer 71 views ### Functional equation and elliptic functions I'm sorry if this question is similar to a one already asked, I'm not aware of many basic facts about elliptic curves. Let g be a meromorphic function on \mathbb{C}, and \Lambda be a lattice in ... 0 votes 0 answers 37 views ### Proof that \frac{1}{2\pi\operatorname{Im}(\tau)}\sqrt{\frac{J(\tau)}{J(\tau)-1}} is holomorphic for \operatorname{Im}(\tau)\gt \frac{5}{4} Let q=e^{2\pi i\tau} and$$E_2(\tau)=1-24\sum_{n=1}^\infty n\frac{q^n}{1-q^n},E_4(\tau)=1+240\sum_{n=1}^\infty n^3\frac{q^n}{1-q^n},E_6(\tau)=1-504\sum_{n=1}^\infty n^5\frac{q^n}{1-q^n},$$... 0 votes 1 answer 73 views ### Understanding a proof of \eta_1\omega_2-\eta_2\omega_1=2\pi i The Weierstrass zeta function \zeta of a lattice L is defined by$$\zeta (z,L)=\frac{1}{z}+\sum_{\omega\in L\setminus\{0\}}\left(\frac{1}{z-\omega}+\frac{1}{\omega}+\frac{z}{\omega^2}\right)$$and ... 4 votes 1 answer 126 views ### Are these sources wrong about Jacobi elliptic functions, and can we fix the flaw? I first noticed that three Youtube videos    give the same definitions of the Jacobi elliptic functions sn, cn, dn (see below), which are different from Wikipedia's. Then I noticed the links ... 2 votes 2 answers 141 views ### Express \sum_{n\in\mathbb{Z}} \left ( q^{(8n+1)^2}-q^{(8n+3)^2}\right ) How can we find a expression for the following sum$$ S=\sum_{n\in\mathbb{Z}} \left ( q^{(8n+1)^2}-q^{(8n+3)^2}\right ) $$where q=e^{-\pi{K^\prime(k)}/{K(k)}} and K(x)=\int_{0}^{1} \frac{1}{\sqrt{... -2 votes 1 answer 87 views ### Deducing the relation f(x)-f(x+1)+f(x+2)-\cdots = 0.5f(x)+Af'(x)+Bf''(x)+\cdots from Abel's "Studies on Elliptic Functions" In Abel's famous article "Studies on Elliptic Functions", most of which I've understood, there is a formula that confused me. It is like this:$$f(x)-f(x+1)+f(x+2)-\cdots = 0.5f(x)+Af'(x)+Bf'... 0 votes 1 answer 61 views ### If$f$is a non-constant elliptic function, then$z\mapsto f'(z)/f(z)$is a non-constant elliptic function How can I prove If$f$is a non-constant elliptic function, then$z\mapsto f'(z)/f(z)$is a non-constant elliptic function? To prove that$z\mapsto f'(z)/f(z)$is elliptic, it suffices to notice ... 0 votes 0 answers 21 views ### Arc length of a graph of an elliptic function Exercise 2 on p. 190 of Greenhill's ''Applications of Elliptic Functions'' asks me to rectify, by means of elliptic arcs,$y=\sin x$and$y=\operatorname{sn}(x,k).$For$y=\sin x$, I get $$\int \... 13 votes 1 answer 353 views ### Hyperbolic sums S(n,k)=\sum_{m=1}^{\infty} \frac{1}{\cosh(\pi m)^{n} \sinh(\pi m)^{k}} It can be verified that$$ \sum_{n=1}^{\infty}\frac{1}{\cosh(\pi n)^3\sinh(\pi n)^2} =\frac{11}{12}-\frac{3K}{2\pi}+\frac{K^2}{2\pi^2}-\frac{K^3}{\pi^3}$$where$K=\frac{\Gamma\left ( \frac14 \right )^...
I have been reading this paper and in the beginning of section 2 it is written something that I am failing to see why it holds. It is stated that if $A$ is a $dxd$ bounded, symmetric matrix such that ...