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- ...are used more often in practical problems than the [[Weierstrass elliptic functions]] as they do not require notions of complex analysis to be defined and/or u ...denoted by pq(u,m), where p and q are any of the letters c, s, n, and d. (Functions of the form pp(u,m) are trivially set to unity for notational completeness. ...42 KB (6,752 words) - 21:05, 28 October 2020
Page text matches
- == Jacobi elliptic functions == ...2 KB (201 words) - 14:35, 1 September 2021
- ...It has a square period lattice and is closely related to the [[Weierstrass elliptic function]] when the Weierstrass invariants satisfy {{math|''g''<sub>2</sub =={{anchor|sl|cl}}Lemniscate sine and cosine functions== ...4 KB (590 words) - 16:05, 3 March 2020
- ...e branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is d ===Other elliptic functions=== ...7 KB (965 words) - 01:00, 9 September 2019
- ...] and [[Leonhard Euler]] ({{Circa|1750}}). Modern mathematics defines an "elliptic integral" as any [[function (mathematics)|function]] {{math|''f''}} which c ...nctions and the three [[Legendre form|Legendre canonical form]]s (i.e. the elliptic integrals of the first, second and third kind). ...21 KB (3,179 words) - 08:59, 16 November 2020
- ...od of [[separation of variables]] applied to the [[Laplace equation]] in [[elliptic coordinates]]. In some special cases solutions can be expressed in terms of ...> the elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of ''B'' the solutions ...9 KB (1,317 words) - 22:17, 1 November 2020
- ...are used more often in practical problems than the [[Weierstrass elliptic functions]] as they do not require notions of complex analysis to be defined and/or u ...denoted by pq(u,m), where p and q are any of the letters c, s, n, and d. (Functions of the form pp(u,m) are trivially set to unity for notational completeness. ...42 KB (6,752 words) - 21:05, 28 October 2020
- {{short description|Special functions of several complex variables}} {{for|other θ functions|Theta function (disambiguation)}} ...28 KB (4,239 words) - 06:08, 1 October 2020
- ...F. Tricomi [http://apps.nrbook.com/bateman/Vol3.pdf Higher Transcendental functions vol. 3] (McGraw Hill, NY, 1953). *{{dlmf|id=31|title=Heun functions|first=B. D.|last=Sleeman|first2=V. B. |last2=Kuznetzov}} ...5 KB (642 words) - 13:31, 14 July 2019
- ...the Appell double series entail [[recurrence relation]]s among contiguous functions. For example, a basic set of such relations for Appell's ''F''<sub>1</sub> ...in terms of [[multiple integral|double integral]]s involving [[elementary functions]] only {{harv|Gradshteyn|Ryzhik|2015|loc=§9.184}}. However, {{harvs | txt | ...15 KB (2,536 words) - 14:44, 16 September 2020
- In [[mathematics]], '''Mathieu functions''', sometimes called angular Mathieu functions, are solutions of Mathieu's [[differential equation]] ...[[partial differential equation]] [[boundary value problem]]s possessing [[elliptic]]{{Disambiguation needed|date=August 2020}} symmetry.<ref name="Gutiérrez-V ...41 KB (6,390 words) - 16:44, 19 November 2020
- ...of [[generalized hypergeometric series]], and are in turn generalized by [[elliptic hypergeometric series]]. ...939-02-06649-2.pdf A q-series identity and the Arithmetic of Hurwitz Zeta Functions]'', (2003) Proceedings of the [[American Mathematical Society]] '''131''', ...11 KB (1,637 words) - 11:47, 27 September 2020
- ...s to the [[generalized hypergeometric function]]. For other hypergeometric functions see [[#See also|See also]].}} ...ented by the '''hypergeometric series''', that includes many other special functions as [[special case|specific]] or [[limiting case (mathematics)|limiting case ...37 KB (5,507 words) - 22:23, 19 October 2020
- ...are poles), but which are not generally solvable in terms of [[elementary functions]]. They were discovered by ...be solved explicitly in terms of integration and previously known special functions. [[Charles Émile Picard|Émile Picard]] pointed out that for orders greater ...18 KB (2,538 words) - 04:25, 20 May 2020
- ...ny particular [[special functions]] as special cases, such as [[elementary functions]], [[Bessel function]]s, and the [[orthogonal polynomials|classical orthogo ...irst3=O. I.|last3=Marichev|title=Integrals & Series Volume 3: More Special Functions|publisher=Gordon and Breach|year=1990|pages=439}}</ref> ...33 KB (5,208 words) - 00:42, 1 November 2020
- Consider the Bessel functions of which there are two kinds, the first and the second. Additionally, there These new noun phrase definitions allow us to distinguish special functions and orthogonal polynomials more adequately in contrast to the previously us ...40 KB (5,985 words) - 10:39, 22 March 2023
- "\\compellintEk": "Legendre's complete elliptic integral of the second kind; Example: \\compellintEk@{k}\nWill be translate "\\compellintEk": "Legendre's complete elliptic integral of the second kind; Example: \\compellintEk@{k}\nWill be translate ...245 KB (19,900 words) - 14:58, 1 September 2021
- ...plane]] that, in non-degenerate cases, satisfy the [[Implicit and explicit functions|implicit]] equation<ref>{{cite book|url=https://books.google.com/books?id=y ...th> is the eccentricity, and the function <math>E</math> is the [[complete elliptic integral of the second kind]], ...76 KB (12,397 words) - 07:11, 3 November 2020
- ...popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of [[probability]] and [[statis ...anslated version of the factorial function,<ref>{{cite book |title=Special Functions: A Graduate Text |first1=Richard |last1=Beals |first2=Roderick |last2=Wong ...72 KB (11,210 words) - 04:19, 5 November 2020