Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
EllipticPi




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,m] > Identities > Functional identities





http://functions.wolfram.com/08.03.17.0002.01









  


  








Input Form





EllipticPi[n, m] == (1/(1 - n)) Sqrt[1/(1 - m)] EllipticPi[n/(n - 1), m/(m - 1)] - (1 - Sqrt[1/(1 - m)] Sqrt[1 - m]) ((I m)/(m - n)) EllipticPi[(n (1 - m))/(n - m), 1 - m] + (1/2) (1 - Sqrt[1/(1 - n)] Sqrt[1 - n]) (Pi/(Sqrt[1 - n] Sqrt[1 - m/n]))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]]], SqrtBox[FractionBox["1", RowBox[List["1", "-", "m"]]]], RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", RowBox[List["n", "-", "1"]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", "m"]]]], SqrtBox[RowBox[List["1", "-", "m"]]]]]]], ")"]], FractionBox[RowBox[List["\[ImaginaryI]", " ", "m"]], RowBox[List["m", "-", "n"]]], RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["n", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]]]], RowBox[List["n", "-", "m"]]], ",", RowBox[List["1", "-", "m"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", "n"]]]], SqrtBox[RowBox[List["1", "-", "n"]]]]]]], ")"]], FractionBox["\[Pi]", RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["m", "n"]]]]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> &#63449; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#10072; </mo> <mfrac> <mi> m </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> <mo> &#10072; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticPi </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticPi </ci> <apply> <times /> <ci> n </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> </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[FractionBox[RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", "m"]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", RowBox[List["n", "-", "1"]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]]]], RowBox[List["1", "-", "n"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", "m"]]]], " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "m"]], ")"]], " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["n", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]]]], RowBox[List["n", "-", "m"]]], ",", RowBox[List["1", "-", "m"]]]], "]"]]]], RowBox[List["m", "-", "n"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["1", RowBox[List["1", "-", "n"]]]], " ", SqrtBox[RowBox[List["1", "-", "n"]]]]]]], ")"]], " ", "\[Pi]"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["m", "n"]]]]]], ")"]]]]]]]]]]]


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





2007-05-02