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variants of this functions
EllipticPi




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,z,m] > Differentiation > Symbolic differentiation > With respect to m





http://functions.wolfram.com/08.06.20.0016.01









  


  








Input Form





D[EllipticPi[n, z, m], {m, p}] == ((Sin[z]^(2 p + 1) Pochhammer[1/2, p])/ (2 p + 1)) Sum[((Pochhammer[1/2 + p, Subscript[k, 1] + Subscript[k, 2] + Subscript[k, 3]] Pochhammer[1, Subscript[k, 1]])/ (Pochhammer[3/2 + p, Subscript[k, 1] + Subscript[k, 2] + Subscript[k, 3]] Subscript[k, 1]! Subscript[k, 2]! Subscript[k, 3]!)) Pochhammer[1/2, Subscript[k, 2]] Pochhammer[1/2 + p, Subscript[k, 3]] n^Subscript[k, 1] Sin[z]^(2 (Subscript[k, 1] + Subscript[k, 2] + Subscript[k, 3])) m^Subscript[k, 3], {Subscript[k, 1], p, Infinity}, {Subscript[k, 2], 0, Infinity}, {Subscript[k, 3], 0, Infinity}] /; Element[p, Integers] && p >= 0








Standard Form





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MathML Form







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</mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> k </mi> <mn> 3 </mn> </msub> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> <degree> <ci> p </ci> </degree> </bvar> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> p </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <ci> p </ci> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> p </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> p </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> m </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> p </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> p </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02