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# Questions tagged [elliptic-functions]

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I'm a little confused at the sigma orientation of tmf, see e.g. Witten genus and its references. The Weierstrass sigma function can be written as $$\sigma_L(z)(q)=\frac{z}{\exp\left(\sum_{k\ge 2} G_k(... 0answers 59 views ### Evaluate a pair of integrals involving dilogarithms over the unit interval These are two variations on the "Bonus round" problem, expertly address by student at the end of his answer to A pair of integrals involving square roots and inverse trigonometric functions over the ... 2answers 347 views ### Are these 5 the only eta quotients that parameterize x^2+y^2 = 1? Given the Dedekind eta function \eta(\tau), define,$$\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\;$$... 0answers 88 views ### Evaluating a Fermi gas problem for a SO(2N+1) matrix integral I have the following multiple integral derived from a random matrix calculation I wish to evaluate$$\int_0^{\pi} dx_1 dx_2 \cdots dx_n \rho(x_1,x_2)\cdots \rho(x_n,x_1)$$where the \rho functions ... 1answer 69 views ### Behaviour of elliptic functions near degenerate lattice What can be said about elliptic functions (Weierstrass \wp, \sigma, Jacobi \theta, sn, cn, etc.) in the limit of degenerate lattice. By "degenerate" I mean \tau = \omega_3/\omega_1 tends to a ... 2answers 381 views ### Solution of an equation with Jacobi theta function I have been struggling with this equation for some time and I do not seem to find any conclusive answer (it's from my research, not a homework). It has to do with the real solutions x to the ... 2answers 259 views ### Bounding an elliptic-type integral Let K>L>0. I would like to find a good upper bound for the integral$$\int_0^L \sqrt{x \left(1 + \frac{1}{K-x}\right)} \,dx.$$An explicit expression for the antiderivative would have to ... 1answer 80 views ### Limits of a quasiperiodic function with two pseudoperiods Let \beta be a real number such that \beta^2\notin\mathbb{Q}. For any smooth function f on \mathbb{R} that decreases sufficiently at infinity, for example a Gaussian function, let us define$$ ...
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Can anyone help me figure out how the identity below was obtained? $\frac{1}{\sqrt{(e_1-e_3)(e_2-e_3)}} = R \prod \limits_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{... 0answers 53 views ### Will a slightly differently shaped torus make this guess about plane sections of a torus true? Jacobi's elliptic functions and plane sections of a torus After Greg Egan posted an excellent answer to a question of mine, which I accepted, I posted my own answer, linked above. The question ... 5answers 2k views ### The letter$\wp$; Name & origin? Do you think the letter$\wp$has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English ... 2answers 243 views ### Differentiating the inverse Weierstrass P-function I will begin with some background: The solutions$\theta$of $$\cos \theta=x$$ constitute of two families, each of which is an arithmetic progression. Namely, if$\arccos x$denotes any particular ... 2answers 363 views ### Expression for infinite product can anyone show me how $$\displaystyle\frac{4}{R}\displaystyle\Pi_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\... 1answer 342 views ### Jacobi and Weierstrass elliptic function Jacobi elliptic function \mathrm{sn} is defined as$$\operatorname{sn}(u,k)=x\Leftrightarrow u=\int_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.$$and Weierstrass sigma function \sigma is defined as ... 2answers 356 views ### Special values of the modular J invariant A special value:$$ 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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