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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 on branch cuts > Formulas on real axis for real m, n > For n<1<m,Pi(u+1/2)<xu+1)-csc-1(m1/2)/;uZ





http://functions.wolfram.com/08.06.06.0035.01









  


  










Input Form





EllipticPi[n, z, m] \[Proportional] Exp[(-Pi) I Floor[Arg[z - x]/(2 Pi)]] EllipticPi[n, x, m] + ((-(1/Sqrt[m])) EllipticPi[n/m, 1/m] + 2 (Floor[x/Pi - 1/2] + 1) EllipticPi[n, m]) (1 - Exp[(-Pi) I Floor[Arg[z - x]/(2 Pi)]]) + Exp[(-Pi) I Floor[Arg[z - x]/(2 Pi)]] ((1/(Sqrt[1 - m Sin[x]^2] (1 - n Sin[x]^2))) (z - x) - ((Sin[2 x] (-m - 2 n + 3 m n Sin[x]^2))/(4 (1 - m Sin[x]^2)^(3/2) (n Sin[x]^2 - 1)^2)) (z - x)^2 + \[Ellipsis]) /; (z -> x) && Element[x, Reals] && Element[m, Reals] && Element[n, Reals] && n < 1 < m && Pi/2 + Pi u < x < Pi (u + 1) - ArcCsc[Sqrt[m]] && Element[u, Integers]










Standard Form





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MathML Form







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<times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticPi </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> <apply> 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<apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> x </ci> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <ci> u </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccsc /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> u </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





© 1998- Wolfram Research, Inc.