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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 at m==1





http://functions.wolfram.com/08.06.06.0089.01









  


  










Input Form





EllipticPi[n, z, m] \[Proportional] (Sqrt[n] Log[(1 + Sqrt[n] Sin[z])/(1 - Sqrt[n] Sin[z])] - 2 Log[Sec[z] + Tan[z]])/(2 (n - 1)) + (Sin[z]^3/6) AppellF1[3/2, 2, 1, 5/2, Sin[z]^2, n Sin[z]^2] (m - 1) + ((3 Sin[z]^5)/40) AppellF1[5/2, 3, 1, 7/2, Sin[z]^2, n Sin[z]^2] (m - 1)^2 + \[Ellipsis] /; (m -> 1) && Abs[Re[z]] <= Pi/2










Standard Form





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MathML Form







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</mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8804; </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sin /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sin /> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <apply> <plus /> <apply> <sec /> <ci> z </ci> </apply> <apply> <tan /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ci> AppellF1 </ci> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='integer'> 2 </cn> <cn type='integer'> 1 </cn> <cn type='rational'> 5 <sep /> 2 </cn> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <cn type='integer'> 40 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> AppellF1 </ci> <cn type='rational'> 5 <sep /> 2 </cn> <cn type='integer'> 3 </cn> <cn type='integer'> 1 </cn> <cn type='rational'> 7 <sep /> 2 </cn> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List[SqrtBox["n"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["1", "+", RowBox[List[SqrtBox["n"], " ", RowBox[List["Sin", "[", "z", "]"]]]]]], RowBox[List["1", "-", RowBox[List[SqrtBox["n"], " ", RowBox[List["Sin", "[", "z", "]"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["Sec", "[", "z", "]"]], "+", RowBox[List["Tan", "[", "z", "]"]]]], "]"]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]]], "+", RowBox[List[FractionBox["1", "6"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "3"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", "2", ",", "1", ",", FractionBox["5", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "40"], " ", RowBox[List["(", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "5"]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["5", "2"], ",", "3", ",", "1", ",", FractionBox["7", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["m", "\[Rule]", "1"]], ")"]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]]]










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





2007-05-02





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