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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 z==0





http://functions.wolfram.com/08.06.06.0068.01









  


  










Input Form





EllipticPi[n, z, m] \[Proportional] (-((Sqrt[1 + 1/Sqrt[1 - m]] Sqrt[Pi] n)/ (2^(5/2) (((-2 (1 + Sqrt[1 - m]) + m)/m)^(3/2) Sqrt[1 - n])))) Sum[(((-1)^q 2^(2 q + 1))/(2 q + 1)!) Sum[StirlingS2[2 q, k] Sum[(((-1)^j j! Binomial[k, k - j] n^j)/ (2^k Gamma[1/2 - k + j])) (m/(1 - Sqrt[1 - m]))^(k - j) ((1 + Sqrt[1 - n])^(-1 - j) - (1 - Sqrt[1 - n])^(-1 - j)) AppellF1[1/2, 1/2, -(3/2), 1/2 - k + j, 1/2 - 1/(2 Sqrt[1 - m]), (2 (1 + Sqrt[1 - m]))/m] z^(2 q + 1), {j, 0, k}], {k, 0, 2 q}], {q, 0, Infinity}] /; Abs[z] < 1










Standard Form





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MathML Form







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type='integer'> 2 </cn> <apply> <plus /> <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> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










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["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", FractionBox["5", "2"]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SqrtBox[RowBox[List["1", "-", "m"]]]]]]], " ", SqrtBox["\[Pi]"], " ", "n"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["2", " ", "q"]]], RowBox[List[RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["2", " ", "q"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["j", "!"]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", SuperscriptBox["n", "j"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["m", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "m"]]]]]], ")"]], RowBox[List["k", "-", "j"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "n"]]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "n"]]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["3", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "+", "j"]], ",", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]]]], ",", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "m"]]]]], ")"]]]], "m"]]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]]]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k", "+", "j"]], "]"]]]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "q"]], "+", "1"]], ")"]], "!"]]]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "m"]]]]], ")"]]]], "+", "m"]], "m"], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List["1", "-", "n"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]]










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





2007-05-02





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