Questions tagged [elliptic-functions]

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

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$f(x)^3+g(x)^3+h(x)^3=1$ relation with elliptic integrals

$x\in\mathbb{C}$ and $f,g,h:\mathbb{C}\to\mathbb{C}$ $f,g,h$ are Meromorphic functions. $$f(x)^3+g(x)^3+h(x)^3=1 \tag{1}$$ $$f'(x)=g(x)^2-h(x)^2 \tag{2}$$ $$g'(x)=h(x)^2-f(x)^2 \tag{3}$$ Initial ...
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Prove an infinite product factorization of lemniscate sine

I recently read about the lemniscate sine function. The function $sl$ is defined as the inverse of $\mathrm{arcsl}(x)=\int_0^x \frac{\mathrm{d}t}{\sqrt{1-t^4}}$. We know that it is an elliptic ...
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44 views

Derivative of Jacobi elliptic function

Let $k \in (0,1)$ and $w>0$. Consider the function $\varphi: \mathbb{R} \longrightarrow \mathbb{R}$ given by $$\varphi(\xi)= \frac{\sqrt{2}k}{\sqrt{k^2+1}}\cdot \text{sn} \left(\frac{\xi}{\sqrt{w}\...
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27 views

Weierstrass Elliptic functions

Please suggest a good book on preliminary Weierstrss elliptic functions or some link from where I can learn about them in details. Please help.
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How to use Lemniscate sine and Lemniscate cosine elliptic integrals?

I’ve been reading up on lemniscate sine and cosine functions and found that they are essentially the lemniscate analogues of circular sine and cosine functions. The following wiki page elaborates a ...
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Extension of conformal mapping of parallelogram onto a half-plane to an elliptic function?

Let $D$ be the interior of a closed parallelogram in the complex plane with one of its vertices at the origin, and $f(z)$ be a conformal mapping of $D$ onto a half-plane $H$ ($f$ is a bijection). ...
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30 views

Position of point which has moved a stated length along an ellipse

Suppose I have some ellipse described by the equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Which is described by the function $f(x)$ in the upper-half plane. Say I move a point from $(0,f(0))$ ...
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138 views

Integrating an ODE in terms of elliptic functions

Consider the ODE for $w:\mathbb{C}\to \mathbb{C}$ \begin{align} w''=2w^3+Aw+B &&(1) \end{align} We can multiply it by $w'$ and then integrate to get \begin{align} w'^2=w^4+Aw^2+2Bw+C &&...
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How can i prove that the elliptic intergal is equal to this double sum? $\Pi(n|m)=\int _0^{\frac{\pi}{2}}\frac{1}{(1-n\sin^2t)\sqrt{1-m\sin^2t}}dt=..$

How can i show that those are the same? $\Pi \left(n|m\right)=\int _0^{\frac{\pi }{2}}\frac{1}{\left(1-n\sin ^2t\right)\sqrt{1-m\sin ^2t}}dt=\frac{\pi }{2}\sum _{k=0}^{\infty }\sum _{j=0}^k\frac{\left(...
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How to show the “naive” Weierstrass elliptic function does not converge absolutely

Several resources (e.g., Stein and Shakarchi, Complex Analysis) begin a discussion of the Weierstrass $\wp$ function by saying that, in order to construct a doubly periodic meromorphic function with ...
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Finite analog of these two infinite series with inverse tangents?

The identity $$ \sum_{n=1}^{\infty}\chi(n)\arctan e^{-\alpha n}+\sum_{n=1}^{\infty}\chi(n)\arctan e^{-\beta n}=\frac{\pi}{8}, \qquad \alpha\beta=\frac{\pi^2}{4},\tag{1} $$ where $\chi(n)=\sin\frac{\...
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Weierstrass elliptic function identity

For a lattice $\Lambda = [\lambda_1, \lambda_2] \subset \mathbb C$, the Weierstrass $\wp$-function defined as \begin{equation} \wp(z) = \frac{1}{z^2} + \sum_{\lambda \in \Lambda \setminus \{0\}} \...
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1answer
51 views

Relation between Weierstrass $\wp$-functions

Let $\Lambda=[\lambda_1,\lambda_2]$ be a lattice with associated Weierstrass function $\wp$, and consider the Weierstrass function $\wp_2$ associated to the lattice $\Lambda_2=[\tfrac{1}{2}\lambda_1,\...
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Jacobian elliptic functions with complex modulus

Let $\mathrm{sn}(u,k)$ be the usual Jacobian elliptic function defined in terms of theta functions: $$\mathrm{sn}(u,k) = \frac{\theta_3(0)}{\theta_2(0)} \frac{\theta_1(z)}{\theta_4(z)}$$ where $z = u/\...
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Relation between different types of elliptic functions

Disclamer: all formulas in this post are intended to be schematic, coefficients signs etc. are mostly wrong. I do various physics-motivated computations involving elliptic functions and the more ...
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Can I bring this integral into a form involving known functions?

In the context of quantum field theory, I am facing the following $1$-dimensional integral over Feynman parameters: $$I(a,b) = \int_0^\infty d\alpha \frac{1}{F G} \arctan \frac{F}{G} \tag{1}$$ with ...
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1answer
120 views

can this integral be expressed in elementary functions?

I have been trying to find the length of an arc of an ellipse and I have been stuck with this integral for a complete day : $$\int_{0}^{x} \sqrt{a^2\cos^2t+b^2\sin^2t} dt$$ And my question is : can ...
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Finding Points on a curve, that have a specific distance

If you have any function or curve, and a point $(x_0|y_0)$, what formula do you use to determine a point with a specific distance D, on that curve. Let's say, given an ellipse, you need to determine ...
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52 views

Proving surjectivity of the $J$-invariant

I'm trying to understand the following Theorem and proof in Apostol's Modular Functions and Dirichlet Series in Number Theory. The $J$-invariant is defined by $ J(\tau)=\frac{g_2(1,\tau)}{g_2(1,\tau)^...
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Regarding property of odd elliptic functions

While self studying analytic number theory from Tom M Apostol modular functions and Dirichlet series in number theory I am unable to think about an argument which Apostol doesn't proves but uses it in ...
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Doubt in proof of theorem proving analyticity of $\Delta(\tau) $ in H.

While studying Number theory from Apostol's Modular functions and Dirichlet series in number theory I have a doubt in proof of theorem 1.15 . It's image is -> I have doubt in line 11 when Apostol ...
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Regarding a doubt in Proof of Theorem in chapter - Elliptic functions of Apostol modular function and Dirichlet series in number theory

I am self studying Analytic number theory from Tom M Apostol modular functions and Dirichlet Series in Number theory. I have doubt in 7th line of proof . My doubt is how if choice of smallest non -...
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27 views

A result related to elliptic functions

I am trying exercises in elliptic functions from Tom M Apostol and I am unable to think about this problem. Problem is --> Prove that every elliptic function for can be expressed in the form for(...
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46 views

Proving a function to be exponential given that it satisfies certain equations.

I am trying exercises of Tom M Apostol Dirichlet series and Modular functions in number theory and I could not think about this problem which is in chapter Elliptic functions. Problem is - Let $\...
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1answer
54 views

Regarding Periods of Weierstrass $\wp$ function

I am trying problems from Tom M Apostol Modular Functions and Dirichlet Series in number theory and I am struck on this problem which is from Chapter - Elliptic Functions. Problem is - Prove that $...
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52 views

Proving a result regarding period of elliptic functions

I am trying exercise of Ch - 1 of Tom M Apostol Modular functions and Dirichlet series in number theory. I am self studying and I am struck on this problem of Elliptic Functions. Problem is --> ...
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67 views

Elliptic curve with 3 real roots

The following is motivated by Silverman's exercise 6.7. Let $\Lambda = <1,\tau>\subset \mathbb{C}$ be a lattice and define the Weierstrass $\wp$-function to be \begin{align*} \wp(z) = \frac{1}{z^...
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32 views

Applications of two representation theorems of elliptic functions and modular forms

I'd like to know some elementary applications of these theorems, and also some examples of how one could work out the representations in question given a particular $f$. If $f$ is an elliptic ...
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On equivalence of pairs of doubly periodic functions

I am self studying analytic number theory from Tom M Apostol Modular functions and Dirichlet series in number theory. I am struck on proving this theorem and I need help Theorem is - Two pairs ( ...
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Minimizing Disttance Between 2 Points on 2 Distinct Elliptical Orbital Functions

The objective is to find the minimum distance from a point $(A_1, P_1, \phi_1)$ on the first orbit to a point $(A_2,P_2, \phi_2)$ on the second orbit. For a measure of distance we can use the function ...
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Why does PGL(2, C) give conformal maps of C?

I'm studying complex analysis and Möbius transformations are introduced. I understood that those class of mappings are recurring when studying conformal mappings of $\mathbb{C}$ (conformal mappings ...
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51 views

About the connection of the Eisenstein series with the Weierstrass ℘ function

I learned about the connection of the Eisenstein series with the Weierstrass ℘ function. For z near 0, the equality below holds. (Complex Analysis, Stein & Shakarchi, p.274) $$℘(z) = 1/z^2 ...
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52 views

Does anybody know this relation between Jacobi theta functions?

I'm proving a theorem that I know is correct, but I would need to proof that $$ \theta{}_{2}\left(z\right)\theta{}_{3}\left(z\right)=\theta'{}_{4}\left(z\right)\theta{}_{1}\left(z\right)-\theta'{}_{1}\...
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68 views

'Even-order of zeros and poles' property of an even elliptic function

While I am studying that every even elliptic function with periods 1 and τ is a rational function of Weierstrass function from 'Complex Analysis' by Stein and Shakarchi(p.271), I have exactly the same ...
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65 views

Computing a definite integral in terms of elliptic functions

I want to perform a definite integral $ \int_{t_1}^{t_2} \frac{dt}{\sqrt{a t^3+ b t^2+ct+d}}, $ and will be happy to get the answer in terms of elliptic functions. Can somebody please guide me on how ...
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71 views

Why the derivative of the logarithm of a theta function is not an elliptic function?

Let's consider the theta functions periodicity conditions $$\vartheta\left(z+1\right) =\vartheta\left(z\right),$$ $$\vartheta\left(z+\tau\right) =e^{-\pi i\tau-2\pi iz}\vartheta\left(z\right),$$ Those ...
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285 views

How to solve this integral using some special functions? $I=\int_a^b \frac{\sin u}{(c-u-\sin u)^{2/3}}du$

This integral cannot be solved with any standard function $$I=\int_a^b \frac{\sin u}{(c-u-\sin u)^{2/3}}du$$ $c,a,b\in \mathbb{R}^+\;\;,\;\;c>u+\sin u \, \forall\;u\in[a,b]$ It is possible to ...
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54 views

Factorization of elliptic functions with theta functions

I found in Mumford (tata Lectures on Theta) that every elliptic function can be written as fraction of theta functions $$\prod\frac{\vartheta_{a_{j}}\left(z-z_{j}\right)}{\vartheta_{b_{i}}\left(z-z_{...
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71 views

Two Equivalent Equations for the Zeros of the Jacobi Theta Function?

I'm trying to find the 0's for the Jacobi theta function with characteristic: $$ \vartheta_{a,b}(z, \tau) :=\sum^\infty_{n=-\infty} e^{\pi i (n + a)^{2} \tau + 2 \pi i(n + a)(z + b)},\quad a,b \in \...
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Coefficients of the power series solution for $f'(x)^2+f(x)^4=1$

I am attempting to calculate the power series coefficients of the solution to the differential equation $$f'(x)^2+f(x)^4=1\qquad f(0)=0.\tag{1}$$ I am trying to do so, because $f(x)$ is the ...
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How to solve a quintic polynomial using elliptic functions with Mathematica?

I followed the exact steps from this forum post to solve quintic polynomials of the form: $x^5 - x + d$ But I got a different answer in number form from Mathematica than the original quintic ...
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Reference for result on elliptic functions and first order ODE

I've been reading chapter 2 of this and finally found the precise form of a result I've been looking for. Namely: Let $F(u',u,x) = 0$ be a first order ODE where $F$ is a polynomial in $u'$ and $u$, ...
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435 views

What even *are* elliptic functions?

I am just beginning to learn about elliptic functions. Wikipedia defines an elliptic function as a function which is meromorphic on $\Bbb C$, and for which there exist two non-zero complex numbers $\...
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Weierstrass elliptic functions and ordinary differential equations

I am studying Elliptic functions for a University project with a particular focus on Weierstrass's theory. For the past few weeks I have been studying various basic properties of the $\wp$ function (...
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1answer
128 views

Weierstrass elliptic funcion in Laurent series form

Could anyone please help me to figure out how $$f_0(z) = \wp(\log z; i\pi, \log \rho)$$ where $\wp$ denotes the Weierstrass elliptic function and $i \pi$ and $\log \rho$ are its half periods. $$ f_0(...
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Divergence of sum over lattice.

This is follow up to my last question on summing over countably infinite sets. I understand the idea conceptually now but I am still stuck when dealing with a concrete example. Specifically, consider ...
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47 views

Meaning of absolute convergence when summing over countably infinite set.

I'm trying to understand the chapter on Elliptic functions in Stein's Complex Analysis. In particular, I am interested in the construction of Weierstrass's $\wp$ function. Let $\Lambda = \{n + m\tau :...
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37 views

Show that whenever $r \gt 2$ , the series $\sum |n+m\tau|^{-r}$ converges uniformly in every half-plane Im$(\tau)\ge \delta \gt 0$

Show that whenever $r \gt 2$ , the series $$\sum_{(n,m)\neq (0,0),n,m\in Z} |n+m\tau|^{-r}$$ converges uniformly in every half-plane Im$(\tau)\ge \delta \gt 0$ . Note that $$\sum_{(n,m) \neq (0,0), n,...
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48 views

How many inputs does the following theta function have?

In the book http://renaissance.ucsd.edu/courses/mae207/mech.pdf page 118 The theta function contains only 1 input, isn't it suppose to be 2 inputs? how does one of the theta functions in this page ...
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53 views

Is it true $\underset{x\to\infty}{\text{lim}}\left(\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)-x\right)=0$?

Figure (1) below illustrates the Jacobi theta function $\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)$ and the linear function $x$ in orange and blue respectively. Figure (1): Illustration of $\...

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