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Citation bot (talk | contribs) Alter: url, journal. URLs might have been anonymized. Add: bibcode, doi. Removed URL that duplicated identifier. Removed access-date with no URL. | Use this bot. Report bugs. | Suggested by Abductive | Category:Special functions | #UCB_Category 115/143 |
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:<math>\frac{1 - \sqrt[4]{1 - k^2}}{1 + \sqrt[4]{1 - k^2}} = \frac{\vartheta_{00}[q(k)] - \vartheta_{01}[q(k)]}{\vartheta_{00}[q(k)] + \vartheta_{01}[q(k)]} = \biggl[\sum_{n = 1}^{\infty} 2\,q(k)^{(2n - 1)^2}\biggr] \biggl[1 + \sum_{n = 1}^{\infty} 2\,q(k)^{4n^2}\biggr]^{-1}</math>
Since we may reduce to the case where the imaginary part of <math>\tau</math> is greater than or equal to <math>\sqrt{3}/2</math> (see [[Modular group#Relationship to hyperbolic geometry|Modular group]]), we can assume the absolute value of <math>q</math> is less than or equal to <math>\exp(-\pi\sqrt{3}/2) \approx 0.0658 </math>; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for <math>q</math>. By solving this function after q we get:<ref>{{cite web|access-date=2023-05-28|title=A002103 - OEIS|url=https://oeis.org/A002103}}<!-- auto-translated by Module:CS1 translator --></ref><ref>{{cite web|access-date=2023-05-28|language=en|title=Series Expansion of EllipticNomeQ differs from older Mathematica Version|url=https://mathematica.stackexchange.com/questions/269455/series-expansion-of-ellipticnomeq-differs-from-older-mathematica-version}}<!-- auto-translated by Module:CS1 translator --></ref><ref>{{citation
:<math>q(k) = \sum_{n = 1}^{\infty} \frac{\text{Sw}(n)}{2^{4n - 3}} \biggl(\frac{1 - \sqrt[4]{1 - k^2}}{1 + \sqrt[4]{1 - k^2}}\biggr)^{4n - 3} = k^2\biggl\{\frac{1}{2} + \biggl[\sum_{n = 1}^{\infty} \frac{\text{Sw}(n + 1)}{2^{4n + 1}} k^{2n}\biggr]\biggr\}^4</math>
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Yet, another method for a rapidly converging fast computation of the Jacobi elliptic sine function found in the literature is shown below<ref>{{Cite journal |last=Smith |first=John I. |date=May 5, 1971 |title=The Even- and Odd-Mode Capacitance Parameters for Coupled Lines in Suspended Substrate |url=https://ieeexplore.ieee.org
Let:
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