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variants of this functions
EllipticPi






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,m] > Representations through more general functions > Through Meijer G > Classical cases





http://functions.wolfram.com/08.03.26.0002.01









  


  










Input Form





EllipticPi[1 - 1/Sqrt[z], 1 - 1/z] == ((Sqrt[z] + 1)/2) EllipticK[1 - 1/z] - (1/(4 Pi)) (EllipticK[1 - 1/z] MeijerG[{{1/2, 3/2}, {}}, {{0, 1}, {}}, z] + EllipticE[1 - 1/z] MeijerG[{{3/2, 3/2}, {}}, {{1, 1}, {}}, z]) /; !IntervalMemberQ[Interval[{-Infinity, -1}], z]










Standard Form





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MathML Form







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</mo> <mrow> <mi> z </mi> <mo> &#8713; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List[RowBox[List["1", "-", FractionBox["1", SqrtBox["z_"]]]], ",", RowBox[List["1", "-", FractionBox["1", "z_"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SqrtBox["z"], "+", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", FractionBox["1", "z"]]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", FractionBox["1", "z"]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["EllipticE", "[", RowBox[List["1", "-", FractionBox["1", "z"]]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]










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





2001-10-29