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        Questions tagged [elliptic-functions]

        The tag has no usage guidance.

        5
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        2answers
        323 views

        Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$?

        Given the Dedekind eta function $\eta(\tau)$, define, $$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$ $$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...
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        0answers
        84 views

        Evaluating a Fermi gas problem for a SO(2N+1) matrix integral

        I have the following multiple integral derived from a random matrix calculation I wish to evaluate $$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$ where the $\rho$ functions ...
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        0answers
        42 views

        Rewriting an elliptic integral in terms of theta functions

        I wish to demonstrate the following from arXiv:hep-th/9808043v2 (equations (3.2) to (3.4)). This is rewriting the following incomplete elliptic integral of the third kind $ \Phi_{\tilde{h}}(h) = \...
        2
        votes
        1answer
        66 views

        Behaviour of elliptic functions near degenerate lattice

        What can be said about elliptic functions (Weierstrass $\wp$, $\sigma$, Jacobi $\theta$, sn, cn, etc.) in the limit of degenerate lattice. By "degenerate" I mean $\tau = \omega_3/\omega_1$ tends to a ...
        6
        votes
        2answers
        360 views

        Solution of an equation with Jacobi theta function

        I have been struggling with this equation for some time and I do not seem to find any conclusive answer (it's from my research, not a homework). It has to do with the real solutions $x$ to the ...
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        votes
        2answers
        253 views

        Bounding an elliptic-type integral

        Let $K>L>0$. I would like to find a good upper bound for the integral $$\int_0^L \sqrt{x \left(1 + \frac{1}{K-x}\right)} \,dx.$$ An explicit expression for the antiderivative would have to ...
        2
        votes
        1answer
        78 views

        Limits of a quasiperiodic function with two pseudoperiods

        Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define $$ ...
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        1answer
        323 views

        Infinite sum and product associated with the Weierstrass elliptic function [closed]

        Can anyone help me figure out how the identity below was obtained? $ \frac{1}{\sqrt{(e_1-e_3)(e_2-e_3)}} = R \prod \limits_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{...
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        0answers
        53 views

        Will a slightly differently shaped torus make this guess about plane sections of a torus true?

        Jacobi's elliptic functions and plane sections of a torus After Greg Egan posted an excellent answer to a question of mine, which I accepted, I posted my own answer, linked above. The question ...
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        votes
        5answers
        1k views

        The letter $\wp$; Name & origin?

        Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English ...
        4
        votes
        2answers
        215 views

        Differentiating the inverse Weierstrass P-function

        I will begin with some background: The solutions $\theta$ of $$\cos \theta=x $$ constitute of two families, each of which is an arithmetic progression. Namely, if $\arccos x$ denotes any particular ...
        -2
        votes
        2answers
        358 views

        Expression for infinite product

        can anyone show me how $$\displaystyle\frac{4}{R}\displaystyle\Pi_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\...
        3
        votes
        1answer
        333 views

        Jacobi and Weierstrass elliptic function

        Jacobi elliptic function $\mathrm{sn}$ is defined as $$\operatorname{sn}(u,k)=x\Leftrightarrow u=\int_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.$$ and Weierstrass sigma function $\sigma$ is defined as ...
        4
        votes
        2answers
        308 views

        Special values of the modular J invariant

        A special value: $$ J\big(i\sqrt{6}\;\big) = \frac{(14+9\sqrt{2}\;)^3\;(2-\sqrt{2}\;)}{4} \tag{1}$$ I wrote $J(\tau) = j(\tau)/1728$. How up-to-date is the Wikipedia listing of known special values ...
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        2answers
        749 views

        Jacobi's elliptic functions and plane sections of a torus

        In $\mathbb R^3$ with Cartesian coordinates $(x,y,z),$ revolve the circle $(x-\sqrt 2)^2+z^2 =1,\ y=0$ about the $z$-axis. This yields a torus embedded in $3$-space that is conformally equivalent to ...

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