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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Representations through more general functions > Through hypergeometric functions of two variables





http://functions.wolfram.com/08.06.26.0004.01









  


  










Input Form





EllipticPi[n, z, m] == (-((Sqrt[1 + 1/Sqrt[1 - m]] Sqrt[Pi] n)/ (2^(5/2) (((-2 (1 + Sqrt[1 - m]) + m)/m)^(3/2) Sqrt[1 - n])))) Sum[(((-1)^q 2^(2 q + 1))/(2 q + 1)!) Sum[StirlingS2[2 q, k] Sum[(((-1)^j j! Binomial[k, k - j] n^j)/ (2^k Gamma[1/2 - k + j])) (m/(1 - Sqrt[1 - m]))^(k - j) ((1 + Sqrt[1 - n])^(-1 - j) - (1 - Sqrt[1 - n])^(-1 - j)) AppellF1[1/2, 1/2, -(3/2), 1/2 - k + j, 1/2 - 1/(2 Sqrt[1 - m]), (2 (1 + Sqrt[1 - m]))/m] z^(2 q + 1), {j, 0, k}], {k, 0, 2 q}], {q, 0, Infinity}] /; Abs[z] < 1










Standard Form





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MathML Form







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</cn> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2007-05-02