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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 n==1





http://functions.wolfram.com/08.06.06.0014.01









  


  










Input Form





EllipticPi[n, z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (Sin[z] AppellF1[1/2, 3/2, 1/2, 3/2, Sin[z]^2, m Sin[z]^2] + (Sin[z]^3/3) AppellF1[3/2, 5/2, 1/2, 5/2, Sin[z]^2, m Sin[z]^2] (n - 1) + (Sin[z]^5/5) AppellF1[5/2, 7/2, 1/2, 7/2, Sin[z]^2, m Sin[z]^2] (n - 1)^2 + \[Ellipsis]) + Round[Re[z]/Pi] (Pi/(Sqrt[1 - m/n] Sqrt[1 - n]) + (2 EllipticE[m])/(m - 1) + 2 EllipticK[m] + ((2 ((1 + m) EllipticE[m] + (-1 + m) EllipticK[m]))/(3 (-1 + m)^2)) (n - 1) - (2/(15 (-1 + m)^3)) ((-3 - 7 m + 2 m^2) EllipticE[m] - (-3 + 2 m + m^2) EllipticK[m]) (n - 1)^2 + \[Ellipsis]) /; (n -> 1) && !Element[(Pi + 2 Re[z])/(4 Pi), Integers]










Standard Form





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MathML Form







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</ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 3 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> 3 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> InterpretationBox </ci> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <ms> 1 </ms> </apply> <ci> AppellF1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 5 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> 5 </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> InterpretationBox </ci> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <ms> 1 </ms> </apply> <ci> AppellF1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 7 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> FractionBox </ci> <ms> 7 </ms> <ms> 2 </ms> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sin </ms> <ms> 2 </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <ms> &#8230; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#8970; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#960; </ms> </apply> <ms> &#8969; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> m </ms> <ms> n </ms> </apply> </list> </apply> </apply> <apply> <ci> SqrtBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> n </ms> </list> </apply> </apply> </list> </apply> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> E </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> K </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> E </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> K </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 3 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 7 </ms> <ms> m </ms> </list> </apply> <ms> - </ms> <ms> 3 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> E </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> m </ms> <ms> 2 </ms> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> m </ms> </list> </apply> <ms> - </ms> <ms> 3 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> K </ms> <ms> ( </ms> <ms> m </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 15 </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 3 </ms> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <ms> &#8230; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#62754; </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> &#172; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> + </ms> <ms> &#960; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> &#960; </ms> </list> </apply> </apply> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8484; </ms> <apply> <ci> Function </ci> <list /> <integers /> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </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[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "3"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["5", "2"], ",", FractionBox["1", "2"], ",", FractionBox["5", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "5"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "5"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox["7", "2"], ",", FractionBox["1", "2"], ",", FractionBox["7", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["m", "n"]]]], " ", SqrtBox[RowBox[List["1", "-", "n"]]]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["m", "-", "1"]]], "+", RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]], RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["7", " ", "m"]], "+", RowBox[List["2", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "m"]], "+", SuperscriptBox["m", "2"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"]]], RowBox[List["15", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "1"]], ")"]], "&&", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]]]]]]]]]]]]










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





2007-05-02





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