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http://functions.wolfram.com/08.06.06.0007.01
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EllipticPi[n, z, m] == Sqrt[n/((n - 1) (m - n))]
(EllipticF[z, m] (Sqrt[((n - 1) (m - n))/n] + ((2 I Pi)/EllipticK[m])
Sum[(EllipticNomeQ[m]^k/(1 - EllipticNomeQ[m]^(2 k)))
Sin[(k Pi a)/EllipticK[m]], {k, 1, Infinity}]) -
2 I Sum[(EllipticNomeQ[m]^k/(k (1 - EllipticNomeQ[m]^(2 k))))
Sin[(k Pi a)/EllipticK[m]] Sin[(k Pi EllipticF[z, m])/EllipticK[m]],
{k, 1, Infinity}]) /; a == InverseJacobiSN[Sqrt[n/m], m] &&
-1 <= n <= 1 && -1 <= m <= 1 && -(Pi/2) <= z <= Pi/2
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["n", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]]]]]], RowBox[List["(", " ", RowBox[List[RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["(", " ", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]]]], "n"]], " ", "+", RowBox[List[FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]], RowBox[List["EllipticK", "[", "m", "]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "k"], RowBox[List["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["2", " ", "k"]]]]]], RowBox[List["Sin", "[", FractionBox[RowBox[List["k", " ", "\[Pi]", " ", "a"]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]]]]]]]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "k"], RowBox[List["k", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["2", " ", "k"]]]]], ")"]]]]], RowBox[List["Sin", "[", FractionBox[RowBox[List["k", " ", "\[Pi]", " ", "a"]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["k", " ", "\[Pi]", " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["a", "\[Equal]", RowBox[List["InverseJacobiSN", "[", RowBox[List[SqrtBox[FractionBox["n", "m"]], ",", "m"]], "]"]]]], "\[And]", RowBox[List[RowBox[List["-", "1"]], "\[LessEqual]", "n", "\[LessEqual]", "1"]], "\[And]", RowBox[List[RowBox[List["-", "1"]], "\[LessEqual]", "m", "\[LessEqual]", "1"]], "\[And]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "\[LessEqual]", "z", "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mi> n </mi> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <msqrt> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> n </mi> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mi> sin </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> ⩵ </mo> <mrow> <msup> <mi> sn </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mfrac> <mi> n </mi> <mi> m </mi> </mfrac> </msqrt> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ≤ </mo> <mi> n </mi> <mo> ≤ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ≤ </mo> <mi> m </mi> <mo> ≤ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ≤ </mo> <mi> z </mi> <mo> ≤ </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> k </ci> <pi /> <ci> a </ci> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> EllipticNomeQ </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> sin </ci> <apply> <times /> <ci> k </ci> <pi /> <ci> a </ci> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> k </ci> <pi /> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> a </ci> <apply> <ci> InverseJacobiSN </ci> <apply> <power /> <apply> <times /> <ci> n </ci> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> m </ci> </apply> </apply> <apply> <leq /> <cn type='integer'> -1 </cn> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <leq /> <cn type='integer'> -1 </cn> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> z </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[FractionBox["n", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]]]], "n"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "k"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["k", " ", "\[Pi]", " ", "a"]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]]]], RowBox[List["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["2", " ", "k"]]]]]]]]]], RowBox[List["EllipticK", "[", "m", "]"]]]]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], "k"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["k", " ", "\[Pi]", " ", "a"]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["k", " ", "\[Pi]", " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], RowBox[List["EllipticK", "[", "m", "]"]]], "]"]]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["EllipticNomeQ", "[", "m", "]"]], RowBox[List["2", " ", "k"]]]]], ")"]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["a", "\[Equal]", RowBox[List["InverseJacobiSN", "[", RowBox[List[SqrtBox[FractionBox["n", "m"]], ",", "m"]], "]"]]]], "&&", RowBox[List[RowBox[List["-", "1"]], "\[LessEqual]", "n", "\[LessEqual]", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "\[LessEqual]", "m", "\[LessEqual]", "1"]], "&&", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "\[LessEqual]", "z", "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
