Questions on doubly periodic functions, like Jacobi and Weierstrass elliptic functions.
9
votes
1answer
83 views
Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$
I need to find a closed form for these nested definite integrals:
$$I=\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$$
The inner integral can be ...
0
votes
1answer
25 views
General equation of an ellipse in 3D space with respect to cylindrical coordinate systems
The regular ellipse formula in 2D is $x^2/a^2 + y^2/b^2 = 1$ but how can it be transformed into a 3D formula including the parameter of $r, \theta$ and $z$?
40
votes
3answers
508 views
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.
I found a solution by myself 10 hours after I posted it, here it is:
...
3
votes
2answers
53 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
89 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 ...
1
vote
2answers
86 views
General solution of a differential equation $x''+{a^2}x+b^2x^2=0$
How do you derive the general solution of this equation:$$x''+{a^2}x+b^2x^2=0$$where a and b are constants.
Please help me to derive solution thanks a lot.
0
votes
0answers
12 views
Semiperiod of $\wp$
If we let $u$ and $v$ span a lattice of $\mathbb{C}$, then for the associated $\wp$-function: $\wp(\frac{u}{2})$ is algebraic over $\mathbb{Q}(G_k(u,v),k \ge 4)$ because it is a zero of $$4X^3 - ...
1
vote
2answers
39 views
How do you solve an equation involving the Weierstrass p-function?
What $z\in \mathbb{C}$ satisfy
\begin{eqnarray*}
\wp(z) = \wp(nz)
\end{eqnarray*}
if $n\in \mathbb{N}$?
The Weierstrass p-function is defined as
\begin{eqnarray*}
\wp(z)=\frac{1}{z^2}+\sum_{\omega\in ...
0
votes
2answers
26 views
Even elliptic function
If I have an elliptic function $f$ with respect to some lattice $L$, and $f$ has poles in the points of $L$ and nowhere else, does it follow that $f$ is even?
I think so, and my idea was to write $f ...
1
vote
2answers
89 views
Analytic functions can't have more than two periods
Let $f(z)$ be a non-constant analytic function.
Show that $f(z)$ can't have more than two periods.
0
votes
0answers
30 views
References for the conformal equivalence of the space of complex 1-tori and C?
What are some good references with proofs of the conformal equivalence of the space of complex tori and $\mathbb{C}$? So far I only have the book by Jones and Singerman.
5
votes
2answers
130 views
Turning an elliptic curve over C into a complex torus
I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would ...
5
votes
1answer
77 views
Direct proof of the non-zeroness of an Eisenstein series
Question: Can you show directly from its formula that $G_4(i)\neq0$?
Recall that the holomorphic Eisenstein series of weight $2k$ is defined by:
$$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
1
vote
0answers
31 views
Question about constructing the Weierstrass $\wp$ function [duplicate]
Possible Duplicate:
convergence of a particular series
I am reading about the construction of the Weierstrass $\wp$-function for an arbitrary lattice in $\mathbb{C}$. If $a, b$ are complex ...
0
votes
0answers
64 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:
...
5
votes
1answer
125 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
137 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 ...
4
votes
0answers
108 views
Inversion of elliptic integral
I have an equation of the type
$$
p=\int_0^b\sqrt{\left(a^2-x^2\right)\left(b^2-x^2\right)}dx,
$$
in which $a$ and $b$ (with $a>b>0$) are (known) functions of some parameter $H$ (such that it is ...
4
votes
2answers
114 views
Klein's j-invariant and Ford circles
Klein's j-invariant has structure which seems to resemble Ford circles:
The latter show up all over number theory (continued fractions, Rademacher's expansion for p(n), etc.)
Can someone explain ...
1
vote
0answers
63 views
$\int_{\gamma}\frac{dz}{\sqrt{1-z^2}}=2\pi$ Along the path $\gamma(t)=2e^{it}$ for $0\leq t\leq 2\pi$ Implies Sine is $2\pi$ periodic
Ok so to back up a bit, by a trig substitution we have for $f:(-1,1)\rightarrow\mathbb{R}$:
$$f(x)=\int_0^x\frac{dt}{\sqrt{1-t^2}} = \arcsin(x)$$
Now according to the notes here: Elliptic Functions ...
2
votes
1answer
85 views
Complex tori as elliptic curves
I have a question about the proof of the following theorem:
A complex torus is conformally equivalent (so isomorphic as Riemann surface) to a complex elliptic curve
I used the book "N.Koblitz, ...
2
votes
1answer
72 views
Functional equation of elliptic function with exactly two poles of order one
This is exercise I.2.2 of "Elliptische Funktionen und Modulformen" by Koecher and Krieg.
Let $f(z)$ be an elliptic function that has exactly two poles of order one at $a$ and $b$ in a fundamental ...
3
votes
0answers
101 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
75 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 ...
5
votes
2answers
167 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
1answer
186 views
Elliptic functions and (meromorphic) simply periodic functions.
Let be $f:\mathbb C\rightarrow\overline {\mathbb C}$ a meromorphic function. The set of periods $\Omega_f$ is a discrete (additive) group and we have one of these possibilities:
i) $\Omega_f= \{0\}$
...
14
votes
1answer
238 views
The importance of modular forms
I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called ...
4
votes
3answers
201 views
Closed form solutions of $\ddot x(t)-x(t)^n=0$
Given the ODE:
$$\ddot x(t)-x(t)=0$$
the solution is:
$$x(t)=C_1\exp(-t)+C_2\exp(t)$$
If we square the $x(t)$ we have:
$$\ddot x(t)-x(t)^2=0$$
and the solution is given by:
$$x(t)=6\wp(t+C_1;0,C_2)$$
...
3
votes
0answers
101 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 ...
0
votes
0answers
78 views
Why is Every Elliptic Function of Order $2$ the Möbius Tranform of a $\wp$-function?
I'm trying to prove that every elliptic function of order $2$ has the form
$$f(z)=\frac{a\wp(z-z_0)+b}{c\wp(z-z_0)+d}$$
I've got the following so far. Let $f$ be an elliptic function of order 2. ...
3
votes
0answers
97 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 ...
3
votes
1answer
209 views
An addition property of Weierstrass $\wp$
I want to show
$$
\left( \begin{array}{ccccc}
&1 &\wp(v) &\wp'(v) \\
&1 &\wp(w) &\wp'(w) \\
&1 &\wp(v+w) &-\wp'(v+w) \end{array} \right)=0
$$
...
9
votes
0answers
160 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 ...
1
vote
3answers
377 views
What does | mean?
I found this symbol on Wolfram|Alpha. Does it mean "or"?
$\displaystyle \large \cos^{-1}(-1)=\mathrm{cd}^{-1}(-1\mid 0)$
5
votes
2answers
188 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
97 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
677 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 ...
3
votes
2answers
306 views
How to calculate $\int \sqrt{(\cos{x})^2-a^2} \, dx$
How to calculate:
$$\int \sqrt{(\cos{x})^2-a^2} \, dx$$
10
votes
1answer
316 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 ...
17
votes
0answers
392 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 ...
14
votes
2answers
464 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 ...
4
votes
1answer
622 views
A Torus and the Weierstrass P function?
In case you don't know what I'm talking about:
http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions
I'm reading Farkas and Kra's "Riemann Surfaces," and they talk briefly about these guys ...
