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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,m] > Series representations > Generalized power series > Expansions at m==infinity





http://functions.wolfram.com/08.03.06.0046.01









  


  










Input Form





EllipticPi[n, m] \[Proportional] (Log[-m]/(2 Sqrt[-m])) (1 + (1 + 2 n)/(4 m) + (3 (3 + 4 n + 8 n^2))/ (64 m^2) + \[Ellipsis]) + ((Sqrt[n] ArcSin[Sqrt[n]])/ (Sqrt[1 - n] Sqrt[-m])) (1 + n/(2 m) + (3 n^2)/(8 m^2) + \[Ellipsis]) + (1/(2 Sqrt[-m])) (4 Log[2] + (-1 + 2 Log[2] + n (-1 + 4 Log[2]))/(2 m) - (21 + 26 n + 28 n^2 - 12 (3 + 4 n + 8 n^2) Log[2])/(64 m^2) + \[Ellipsis]) /; (Abs[m] -> Infinity)










Standard Form





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MathML Form







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</mo> <msqrt> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> n </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> m </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 28 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 26 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 21 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 64 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arcsin /> <apply> <power /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <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> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> m </ci> </apply> <infinity /> </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[FractionBox[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], RowBox[List["4", " ", "m"]]], "+", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "n"]], "+", RowBox[List["8", " ", SuperscriptBox["n", "2"]]]]], ")"]]]], RowBox[List["64", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "m"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["n"], " ", RowBox[List["ArcSin", "[", SqrtBox["n"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["n", RowBox[List["2", " ", "m"]]], "+", FractionBox[RowBox[List["3", " ", SuperscriptBox["n", "2"]]], RowBox[List["8", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", "n"]]], " ", SqrtBox[RowBox[List["-", "m"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["4", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", "2", "]"]]]]]], ")"]]]]]], RowBox[List["2", " ", "m"]]], "-", FractionBox[RowBox[List["21", "+", RowBox[List["26", " ", "n"]], "+", RowBox[List["28", " ", SuperscriptBox["n", "2"]]], "-", RowBox[List["12", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "n"]], "+", RowBox[List["8", " ", SuperscriptBox["n", "2"]]]]], ")"]], " ", RowBox[List["Log", "[", "2", "]"]]]]]], RowBox[List["64", " ", SuperscriptBox["m", "2"]]]], "+", "\[Ellipsis]"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "m"]]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2007-05-02





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