# Questions tagged [elliptic-functions]

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

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### What is the range of the Weierstrass elliptic function?

There is the following function: $$\wp(z,\Lambda):=\frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right)$$ What values can it take? For me ...
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### Minimal polynomial of $\operatorname{cd}\frac{4K}{n}$

Define $$K=\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$ and the Jacobi elliptic function $\operatorname{cd}$ with modulus $i$ by $$\int_{\operatorname{cd}z}^1\dfrac{dx}{\sqrt{1-x^4}}=z$$ on $z\in [0,2K]$ and by ...
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### Given algebraic $a$, find the closed form of $\int_0^a \dfrac{dx}{\sqrt{1-x^4}}$

Let $$A=\int_0^1 \dfrac{dx}{\sqrt{1-x^4}}.$$ Given an algebraic number $0\le a\le 1$, can we determine if there exists a rational number $b$ such that $$\int_0^a \dfrac{dx}{\sqrt{1-x^4}}=Ab?$$ If so, ...
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### Mapping the upper half-plane onto rhombus

Book's question: Map the upper half-plane $\Im z>0$ onto a rhombus in the $w$-plane with angle $\alpha\pi$ at the vertex $A=0$ and side $d$. The correspondence of the points is given by $A=0\to z=0$...
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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 ...