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http://functions.wolfram.com/08.06.26.0003.01
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EllipticPi[n, z, m] == (-1)^Round[Re[z]/Pi] Sin[z]
LauricellaFD[1/2, {1, 1/2, 1/2}, 3/2, {n Sin[z]^2, Sin[z]^2,
m Sin[z]^2}] + 2 Round[Re[z]/Pi] EllipticPi[n, m]
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Cell[BoxData[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["Sin", "[", "z", "]"]], " ", RowBox[List["LauricellaFD", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["{", RowBox[List["1", ",", FractionBox["1", "2"], ",", FractionBox["1", "2"]]], "}"]], ",", FractionBox["3", "2"], ",", RowBox[List["{", RowBox[List[RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "}"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]]]]]
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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> <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> <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> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> F </mi> <mi> D </mi> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mi> n </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> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </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> </mrow> <mo> + </mo> <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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Π </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> ❘ </ms> <ms> m </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ⌊ </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> π </ms> </apply> <ms> ⌉ </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <ms> D </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> 3 </ms> <ms> ) </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> 1 </ms> <ms> , </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </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> <ms> n </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> <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> <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> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> ⌊ </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> π </ms> </apply> <ms> ⌉ </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> Π </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ❘ </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["LauricellaFD", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["{", RowBox[List["1", ",", FractionBox["1", "2"], ",", FractionBox["1", "2"]]], "}"]], ",", FractionBox["3", "2"], ",", RowBox[List["{", RowBox[List[RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "}"]]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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