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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Differentiation > Low-order differentiation > With respect to n





http://functions.wolfram.com/08.06.20.0002.01









  


  










Input Form





D[EllipticPi[n, z, m], {n, 2}] == ((m (1 - 4 n) + n (5 n - 2))/(4 (m - n) (n - 1)^2 n^2)) EllipticF[z, m] - ((m (1 + 2 n) + (2 - 5 n) n)/(4 (m - n)^2 (n - 1)^2 n)) EllipticE[z, m] + ((2 m (2 - 5 n) n + 3 n^4 + m^2 (4 n - 1))/(4 (m - n)^2 (n - 1)^2 n^2)) EllipticPi[n, z, m] + ((Sin[2 z] Sqrt[1 - m Sin[z]^2])/ (16 (m - n)^2 (n - 1)^2 (n Sin[z]^2 - 1)^2)) (n^2 (5 n - 8) - m (n + 2 n^2 - 6) + n (m (1 + 2 n) + (2 - 5 n) n) Cos[2 z])










Standard Form





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MathML Form







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<apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> <cn type='integer'> -8 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <plus /> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> n </ci> <cn type='integer'> -6 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <ci> m </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> </apply> </apply> <ci> n </ci> </apply> <apply> <times /> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> </apply> <apply> <times /> <ci> n </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticE </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.