Jacobi elliptic functions: Difference between revisions

Revision as of 18:32, 23 June 2025

Revision as of 18:36, 23 June 2025

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==Continued fractions==

Assuming real numbers <math>a,p</math> with <math>0<a<p</math> and the [[Nome (mathematics)|nome]] <math>q=e^{\pi i \tau}</math>, <math>\operatorname{Im}(\tau)>0</math> with [[Theta function|elliptic modulus]] <math display="inline">k(\tau)=\sqrt{1-k'(\tau)^2}=(\vartheta_{10}(0;\tau)/\vartheta_{00}(0;\tau))^2</math>. If <math>K[\tau]=K(k(\tau))</math>, where <math>K(x)=\pi/2\cdot {}_2F_1(1/2,1/2;1;x^2)</math> is the [[complete elliptic integral of the first kind]], then holds the following [[Continued fraction|continued fraction expansion]]<ref name="bagis-evaluations">{{citation |mode=cs1 |type=Preprint |first=N. |last=Bagis.(2020)." |year=2018 |title=Evaluations of series related to Jacobi elliptic functions".function preprint https://www.researchgate|arxiv=1803.net/publication/331370071_Evaluations_of_Series_Related_to_Jacobi_Elliptic_Functions09445}}</ref>

:<math>

\begin{align}