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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Series representations > Generalized power series > Expansions at m==1





http://functions.wolfram.com/08.06.06.0091.01









  


  










Input Form





EllipticPi[n, z, m] == Sum[(Pochhammer[1/2, k]/k!) ((n - 1)^(-k - 1) Sqrt[n] ArcTanh[Sqrt[n] Sin[z]] + (1/(2 k!)) Sum[(((-1)^(k - p) Pochhammer[p, 2 (k - p)])/ (2^(2 k - p) (k - p)!)) ((-1)^p p! ((1 - Sqrt[n])^(-1 - p) + (1 + Sqrt[n])^(-1 - p)) ArcTanh[Sin[z]] + Tan[z] Sec[z] Sum[Binomial[p, j] (-1)^(p - j) (p - j)! ((1 - Sqrt[n])^(-1 + j - p) + (1 + Sqrt[n])^(-1 + j - p)) Sum[((2^(1 - j + 2 q) q! Tan[z]^(2 q))/(j - q - 1)!) Pochhammer[2 - j + 2 q, 2 (-1 + j - q)], {q, 0, j - 1}], {j, 0, p}]), {p, 0, k}]) (m - 1)^k, {k, 0, Infinity}] /; Abs[Re[z]] <= Pi/2










Standard Form





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MathML Form







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<apply> <tan /> <ci> z </ci> </apply> <apply> <sec /> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> p </ci> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> q </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> q </ci> </apply> <apply> <power /> <apply> <tan /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <leq /> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





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