4
votes
1answer
52 views

Fundamental period of the WeierstrassP elliptic function?

Consider the WeierstrassP elliptic function $\wp(z, g_2, g_3)$ with the invariants $g_2\in\mathbb{R}$ and $g_3\in\mathbb{R}$: $$\wp'(z)^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3$$ According to Wikipedia when ...
2
votes
0answers
48 views

From inverse Weierstrass function to Jacobi elliptic/inverse elliptic functions?

As a conclusion to a previous question on integrals, I get an answer in terms of inverse Weierstrass elliptic function : $$ f\left(x\right)=\wp^{-1}\left( \beta + \frac{9\beta^2-1}{3(x-\beta)} \right) ...
2
votes
2answers
124 views

Hypergeometric formulas for the j-function

Given the j-function $j(\tau)$ and elliptic lambda function $\lambda(\tau)$. Define, $$g = -1+2\frac{\,_2F_1\big(\tfrac{1}{2},\tfrac{1}{2},1,\,\lambda(2\tau)\big) ...
5
votes
1answer
163 views

How to prove that $\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4)$?

How can we prove that $$\frac{\eta^{14}(q^4)}{\eta^{4}(q^8)}=4\eta^4(q^2)\eta^2(q^4)\eta^4(q^8)+\eta^4(q)\eta^2(q^2)\eta^4(q^4) \ ?$$ Here, $\eta(q)$ is the Dedekind Eta Function, which is defined by ...
3
votes
1answer
120 views

Addition formulas for Jacobi amplitude function

Are there any known summation formulas for the Jacobi amplitude function? I need a formula like $\mathrm{am}(t+x)=\mathrm{am}(t) + f(x)$. I have plotted some graphs and it seems that $f(x)$ is ...
7
votes
3answers
344 views

Can you recommend some books on elliptic function?

I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve?Many thanks in advance!
3
votes
2answers
78 views

Proof of $\left| \operatorname{cn}\left( x(1+ i) \mid m \right)\right|=1$ for $m=\frac{1}{2}$ and $x \in \mathbb{R}$

Let $\operatorname{cn}(u\mid m)$ be Gudermann's notation for the Jacobi elliptic function $\operatorname{cn}$. It is well known to be doubly periodic. For $0<m<1$ the two periods, $4 K(m)$ and ...
1
vote
1answer
258 views

Numerical values of the Jacobi elliptic function sn: Wolfram Alpha vs. Maple vs. C++?

I have a problem to check the validity of an algorithm I've implemented in C++ to compute the Jacobi elliptic function $\mathrm{sn}(u, k)$ (inspired and improved from Numerical Recipes 3rd edition). I ...
5
votes
2answers
345 views

Turning an elliptic curve over C into a complex torus

I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
0
votes
0answers
98 views

Elliptical Integrals and graphing plot

I'm trying to computer plot the graphs of sn(u), cn(u) and dn(u) for k = 1/4, 1/2, 3/4, 0.9 and 0.99 And I am trying to plot 3D graphs of sn, cn and dn as functions of u and k. Here's what I have: ...
7
votes
1answer
162 views

Finding x in $\frac{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,1-x)}{\,_2F_1(\frac{1}{5},\frac{4}{5},\,1,\,x)} = \sqrt{n}$

I was trying to find a closed-form for $0<x<1$ in, $$\frac{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,1-x)}{\,_2F_1(\frac{1}{m},\,1-\frac{1}{m},\,1,\,x)} = \sqrt{n}$$ where $\,_2F_1(a,b,c,z)$ ...
2
votes
1answer
290 views

Conformal mapping from triangle to upper half plane in terms of Weierstrass $\wp$

I'm trying to explicitly compute a conformal map $f:\Delta \rightarrow \mathbb{H}$ where $\Delta$ is a triangle and $\mathbb{H}$ is the upper half plane, in terms of the Weierstrass $\wp$ function. I ...
3
votes
0answers
209 views

Properties of the lemniscate functions as meromorphic functions on $\mathbb{C}$

We consider the following function. $$u(x) = \int_{0}^{x} \frac{dt}{\sqrt{1 - t^4}}$$ $u(x)$ is defined on $[-1, 1]$. Since $u'(x) = \frac{1}{\sqrt{1 - x^4}} > 0$ on $(-1, 1)$, $u(x)$ is strctly ...
1
vote
1answer
140 views

Power series expansion of the lemniscate function

We consider the following function. $$u(x) = \int_{0}^{x} \frac{dt}{\sqrt{1 - t^4}}$$ $u(x)$ is defined on $[-1, 1]$. Since $u'(x) = \frac{1}{\sqrt{1 - x^4}} > 0$ on $(-1, 1)$, $u(x)$ is strctly ...
6
votes
2answers
255 views

Deriving the addition formula for the lemniscate functions from a total differential equation

The lemniscate of Bernoulli $C$ is a plane curve defined as follows. Let $a > 0$ be a real number. Let $F_1 = (a, 0)$ and $F_2 = (-a, 0)$ be two points of $\mathbb{R}^2$. Let $C = \{P \in ...
4
votes
0answers
138 views

Weierstrass $\wp$-Function Addition Property

Consider the function $$ \det\left( \begin{array}{ccccc} &1 &\wp(z) &\wp'(z) \\ &1 &\wp(w) &\wp'(w) \\ &1 &\wp(-z-w) &\wp'(-z-w) \end{array} \right)=f(z) $$ I'm ...
3
votes
0answers
151 views

Equivalent Definitions of the Weierstass $\wp$-Function

I've come across two equivalent definitions of the Weierstrass $\wp$-function, but don't know how to prove that they are equivalent. Definition 1 $\wp(z)=cf(z)+d$ where $f$ is the elliptic function ...
4
votes
1answer
346 views

An addition property of Weierstrass $\wp$

I want to show $$ \left( \begin{array}{ccccc} &1 &\wp(v) &\wp'(v) \\ &1 &\wp(w) &\wp'(w) \\ &1 &\wp(v+w) &-\wp'(v+w) \end{array} \right)=0 $$ ...
13
votes
1answer
270 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
2
votes
3answers
485 views

What does | mean?

I found this symbol on Wolfram|Alpha. Does it mean "or"? $\displaystyle \large \cos^{-1}(-1)=\mathrm{cd}^{-1}(-1\mid 0)$
6
votes
2answers
429 views

Weierstrass $\wp$ function doubly periodic

I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough ...
4
votes
2answers
108 views

Limits of a function involving $\mathrm{cn}(x,k)$

Given $$f(x) = \frac{1 - \mathrm{cn}(x,k)}{{\sqrt3}(1+\mathrm{cn}(x,k)) - 1 + \mathrm{cn}(x,k)}$$ what would be $$\lim_{x\to 0} f(x)$$ and $$\lim_{x\to\infty} f(x)$$ when ...
8
votes
2answers
1k views

Addition theorems for elliptic functions: is there a painless way?

The Weierstrass $\wp$ function satisfies the addition formula $$\wp(z+Z)+\wp(z)+\wp(Z) = \left(\frac{\wp'(z)-\wp'(Z)}{\wp(z)-\wp(Z)}\right)^2.$$ Of course, this is just the $x$-coordinate of ...
11
votes
1answer
387 views

elliptic generalizations of Euler's trick

So Euler employed the following identity $$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$ to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$ I'm curious if there's been ...
26
votes
1answer
609 views

elliptic functions on the 17 wallpaper groups

In doubly periodic functions as tessellations (other than parallelograms), we learned about the Dixonian elliptic functions. There are 17 wallpaper groups -- are there elliptic function analogues for ...
17
votes
2answers
735 views

doubly periodic functions as tessellations (other than parallelograms)

I think of a snapshot of a single period of a doubly periodic function as one parallelogram-shaped tile in a tessellation, could a function have a period that repeats like honeycomb or some other not ...