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http://functions.wolfram.com/08.06.06.0015.01
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EllipticPi[n, z, m] == (-1)^Round[Re[z]/Pi]
Sum[(Sin[z]^(1 + 2 k)/(1 + 2 k)) AppellF1[1/2 + k, 3/2 + k, 1/2, 3/2 + k,
Sin[z]^2, m Sin[z]^2] (n - 1)^k, {k, 0, Infinity}] +
2 Round[Re[z]/Pi] EllipticPi[n, m] /;
!Element[(Pi + 2 Re[z])/(4 Pi), Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]], RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "k"]]]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]], "]"]]]]]]
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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> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ⌉ </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ⌉ </mo> </mrow> </msup> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> sin </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ¬ </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ⌉ </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Π </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ⌉ </mo> </mrow> </msup> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> sin </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ¬ </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> </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[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "k"]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "k"]]], RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]], "/;", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
