Questions tagged [elliptic-functions]
Questions on doubly periodic functions on the complex plane such as Jacobi and Weierstrass elliptic functions.
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questions
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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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12 views
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 ...
0
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1answer
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
$$\varphi(\xi)= \frac{\sqrt{2}k}{\sqrt{k^2+1}}\cdot \text{sn} \left(\frac{\xi}{\sqrt{w}\...
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1answer
27 views
Weierstrass Elliptic functions
Please suggest a good book on preliminary Weierstrss elliptic functions or some link from where I can learn about them in details. Please help.
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1answer
39 views
How to use Lemniscate sine and Lemniscate cosine elliptic integrals?
I’ve been reading up on lemniscate sine and cosine functions and found that they are essentially the lemniscate analogues of circular sine and cosine functions. The following wiki page elaborates a ...
2
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2answers
63 views
Extension of conformal mapping of parallelogram onto a half-plane to an elliptic function?
Let $D$ be the interior of a closed parallelogram in the complex plane with one of its vertices at the origin, and $f(z)$ be a conformal mapping of $D$ onto a half-plane $H$ ($f$ is a bijection). ...
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30 views
Position of point which has moved a stated length along an ellipse
Suppose I have some ellipse described by the equation
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
Which is described by the function $f(x)$ in the upper-half plane. Say I move a point from $(0,f(0))$ ...
3
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1answer
138 views
Integrating an ODE in terms of elliptic functions
Consider the ODE for $w:\mathbb{C}\to \mathbb{C}$
\begin{align}
w''=2w^3+Aw+B &&(1)
\end{align}
We can multiply it by $w'$ and then integrate to get
\begin{align}
w'^2=w^4+Aw^2+2Bw+C &&...
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1answer
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How can i prove that the elliptic intergal is equal to this double sum? $\Pi(n|m)=\int _0^{\frac{\pi}{2}}\frac{1}{(1-n\sin^2t)\sqrt{1-m\sin^2t}}dt=..$
How can i show that those are the same?
$\Pi \left(n|m\right)=\int _0^{\frac{\pi }{2}}\frac{1}{\left(1-n\sin ^2t\right)\sqrt{1-m\sin ^2t}}dt=\frac{\pi }{2}\sum _{k=0}^{\infty }\sum _{j=0}^k\frac{\left(...
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How to show the “naive” Weierstrass elliptic function does not converge absolutely
Several resources (e.g., Stein and Shakarchi, Complex Analysis) begin a discussion of the Weierstrass $\wp$ function by saying that, in order to construct a doubly periodic meromorphic function with ...
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Finite analog of these two infinite series with inverse tangents?
The identity
$$
\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}
$$
where $\chi(n)=\sin\frac{\...
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23 views
Weierstrass elliptic function identity
For a lattice $\Lambda = [\lambda_1, \lambda_2] \subset \mathbb C$, the Weierstrass $\wp$-function defined as
\begin{equation}
\wp(z) = \frac{1}{z^2} + \sum_{\lambda \in \Lambda \setminus \{0\}} \...
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1answer
51 views
Relation between Weierstrass $\wp$-functions
Let $\Lambda=[\lambda_1,\lambda_2]$ be a lattice with associated Weierstrass function $\wp$, and consider the Weierstrass function $\wp_2$ associated to the lattice $\Lambda_2=[\tfrac{1}{2}\lambda_1,\...
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2answers
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Jacobian elliptic functions with complex modulus
Let $\mathrm{sn}(u,k)$ be the usual Jacobian elliptic function defined in terms of theta functions: $$\mathrm{sn}(u,k) = \frac{\theta_3(0)}{\theta_2(0)} \frac{\theta_1(z)}{\theta_4(z)}$$ where $z = u/\...
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1answer
40 views
Relation between different types of elliptic functions
Disclamer: all formulas in this post are intended to be schematic, coefficients signs etc. are mostly wrong.
I do various physics-motivated computations involving elliptic functions and the more ...
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0answers
68 views
Can I bring this integral into a form involving known functions?
In the context of quantum field theory, I am facing the following $1$-dimensional integral over Feynman parameters:
$$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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1answer
30 views
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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1answer
27 views
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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1answer
39 views
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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31 views
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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1answer
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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2answers
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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1answer
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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1answer
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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12 views
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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1answer
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
$$ \theta{}_{2}\left(z\right)\theta{}_{3}\left(z\right)=\theta'{}_{4}\left(z\right)\theta{}_{1}\left(z\right)-\theta'{}_{1}\...
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1answer
68 views
'Even-order of zeros and poles' property of an even elliptic function
While I am studying that every even elliptic function with periods 1 and τ is a rational function of Weierstrass function from 'Complex Analysis' by Stein and Shakarchi(p.271), I have exactly the same ...
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2answers
65 views
Computing a definite integral in terms of elliptic functions
I want to perform a definite integral
$
\int_{t_1}^{t_2} \frac{dt}{\sqrt{a t^3+ b t^2+ct+d}},
$
and will be happy to get the answer in terms of elliptic functions. Can somebody please guide me on how ...
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1answer
71 views
Why the derivative of the logarithm of a theta function is not an elliptic function?
Let's consider the theta functions periodicity conditions
$$\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),$$
Those ...
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2answers
285 views
How to solve this integral using some special functions? $I=\int_a^b \frac{\sin u}{(c-u-\sin u)^{2/3}}du$
This integral cannot be solved with any standard function
$$I=\int_a^b \frac{\sin u}{(c-u-\sin u)^{2/3}}du$$
$c,a,b\in \mathbb{R}^+\;\;,\;\;c>u+\sin u \, \forall\;u\in[a,b]$
It is possible to ...
2
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1answer
54 views
Factorization of elliptic functions with theta functions
I found in Mumford (tata Lectures on Theta) that every elliptic function can be written as fraction of theta functions
$$\prod\frac{\vartheta_{a_{j}}\left(z-z_{j}\right)}{\vartheta_{b_{i}}\left(z-z_{...
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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}$$
I am trying to do so, because $f(x)$ is the ...
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1answer
42 views
How to solve a quintic polynomial using elliptic functions with Mathematica?
I followed the exact steps from this forum post to solve quintic polynomials of the form:
$x^5 - x + d$
But I got a different answer in number form from Mathematica than the original quintic ...
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0answers
22 views
Reference for result on elliptic functions and first order ODE
I've been reading chapter 2 of this and finally found the precise form of a result I've been looking for. Namely:
Let $F(u',u,x) = 0$ be a first order ODE where $F$ is a polynomial in $u'$ and $u$, ...
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2answers
435 views
What even *are* elliptic functions?
I am just beginning to learn about elliptic functions. Wikipedia defines an elliptic function as a function which is meromorphic on $\Bbb C$, and for which there exist two non-zero complex numbers $\...
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Weierstrass elliptic functions and ordinary differential equations
I am studying Elliptic functions for a University project with a particular focus on Weierstrass's theory. For the past few weeks I have been studying various basic properties of the $\wp$ function (...
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1answer
128 views
Weierstrass elliptic funcion in Laurent series form
Could anyone please help me to figure out how
$$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(...
3
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2answers
74 views
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 ...
2
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1answer
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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1answer
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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1answer
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 $\...
