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http://functions.wolfram.com/09.47.06.0010.01
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InverseJacobiSD[z, m] == Sum[(((-1)^j Pochhammer[1/2, j]^2 z^(2 j + 1))/
(Pochhammer[3/2, j] j!)) HypergeometricPFQ[{{1/2}, {-j}, {1/2 + j}},
{{}, {1/2 - j}, {3/2 + j}}, 1, z^2] m^j, {j, 0, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["InverseJacobiSD", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], "2"], SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "j"]], "]"]], RowBox[List["j", "!"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List["-", "j"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "+", "j"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "-", "j"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "j"]], "}"]]]], "}"]], ",", "1", ",", SuperscriptBox["z", "2"]]], "]"]], SuperscriptBox["m", "j"]]]]]]]]]
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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> <msup> <mi> sd </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <msup> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> F </mi> <mrow> <mn> 0 </mn> <mo> ⁢ </mo> <mn> 1 </mn> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> ⁢ </mo> <mn> 1 </mn> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mrow> <mtable> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> j </mi> </mrow> <mo> ; </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> j </mi> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> j </mi> </mrow> <mo> ; </mo> </mrow> </mtd> </mtr> </mtable> <mo> ⁢ </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> m </mi> <mi> j </mi> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> sd </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> ❘ </ms> <ms> m </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> ∞ </ms> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <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> <ms> j </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> j </ms> </apply> <ci> Pochhammer </ci> </apply> <ms> 2 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> j </ms> </list> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <ms> j </ms> </apply> <ci> Pochhammer </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> ! 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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiSD", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], "2"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List["-", "j"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "+", "j"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "-", "j"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], "+", "j"]], "}"]]]], "}"]], ",", "1", ",", SuperscriptBox["z", "2"]]], "]"]], " ", SuperscriptBox["m", "j"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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