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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,m] > Series representations > Other series representations





http://functions.wolfram.com/08.03.06.0007.01









  


  










Input Form





EllipticPi[n, m] == Sqrt[n/((m - n) (n - 1))] EllipticK[m] (Sqrt[((m - n) (n - 1))/n] + ((2 I Pi)/EllipticK[m]) Sum[(EllipticNomeQ[m]^k/(1 - EllipticNomeQ[m]^(2 k))) Sin[((k Pi)/EllipticK[m]) InverseJacobiSN[Sqrt[n/m], m]], {k, 1, Infinity}]) /; -1 <= n <= 1 && -1 <= m <= 1










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mfrac> <mi> n </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> sn </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mi> n </mi> <mi> m </mi> </mfrac> </msqrt> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> n </mi> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8804; </mo> <mi> n </mi> <mo> &#8804; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <times /> <ci> k </ci> <pi /> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> InverseJacobiSN </ci> <apply> <power /> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <leq /> <cn type='integer'> -1 </cn> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <leq /> <cn type='integer'> -1 </cn> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[FractionBox["n", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]]]], " ", RowBox[List["EllipticK", "[", "m", "]"]], " ", RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]], "n"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "k"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", " ", "\[Pi]"]], ")"]], " ", RowBox[List["InverseJacobiSN", "[", RowBox[List[SqrtBox[FractionBox["n", "m"]], ",", "m"]], "]"]]]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]]]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["2", " ", "k"]]]]]]]]]], RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "\[LessEqual]", "n", "\[LessEqual]", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "\[LessEqual]", "m", "\[LessEqual]", "1"]]]]]]]]]]










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





2001-10-29





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