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http://functions.wolfram.com/09.48.27.0018.01
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InverseJacobiSN[z, m] ==
((Sqrt[1 - m z^2] JacobiCD[InverseJacobiSN[z, m], m])/Sqrt[1 - z^2])
EllipticF[ArcSin[z], m] /;
!Exists[\[Tau], {Element[\[Tau], Reals], 0 < \[Tau] < 1},
Im[1 - \[Tau]^2 z^2] == 0 && 1 - \[Tau]^2 z^2 < 0 &&
Im[1 - m \[Tau]^2 z^2] == 0 && 1 - m \[Tau]^2 z^2 < 0]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseJacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List["InverseJacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]], RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], ",", "m"]], "]"]]]]]], "/;", " ", RowBox[List["Not", "[", RowBox[List["Exists", "[", RowBox[List["\[Tau]", ",", " ", RowBox[List["{", RowBox[List[RowBox[List["\[Tau]", "\[Element]", "Reals"]], ",", " ", RowBox[List["0", "<", "\[Tau]", "<", "1"]]]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List["1", "-", RowBox[List[SuperscriptBox["\[Tau]", "2"], SuperscriptBox["z", "2"]]]]], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["1", "-", RowBox[List[SuperscriptBox["\[Tau]", "2"], SuperscriptBox["z", "2"]]]]], "<", "0"]], "\[And]", RowBox[List[RowBox[List["Im", "[", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox["\[Tau]", "2"], SuperscriptBox["z", "2"]]]]], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox["\[Tau]", "2"], SuperscriptBox["z", "2"]]]]], "<", "0"]]]]]], "]"]], "]"]]]]]]
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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> <msup> <mi> sn </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> <mfrac> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> cd </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sn </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> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mi> F </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ¬ </mo> <mrow> <msub> <mo> ∃ </mo> <mrow> <mi> τ </mi> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mi> τ </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> , </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> τ </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> τ </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> τ </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> τ </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <msup> <mi> τ </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> InverseJacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> JacobiCD </ci> <apply> <ci> InverseJacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticF </ci> <apply> <arcsin /> <ci> z </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <not /> <apply> <exists /> <bvar> <ci> τ </ci> </bvar> <bvar> <list> <apply> <in /> <ci> τ </ci> <reals /> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> </list> </bvar> <apply> <and /> <apply> <eq /> <apply> <imaginary /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> τ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> τ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <imaginary /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> τ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> τ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseJacobiSN", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["JacobiCD", "[", RowBox[List[RowBox[List["InverseJacobiSN", "[", RowBox[List["z", ",", "m"]], "]"]], ",", "m"]], "]"]]]], ")"]], " ", RowBox[List["EllipticF", "[", RowBox[List[RowBox[List["ArcSin", "[", "z", "]"]], ",", "m"]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]], "/;", RowBox[List["!", RowBox[List[SubscriptBox["\[Exists]", RowBox[List["\[Tau]", ",", RowBox[List["{", RowBox[List[RowBox[List["\[Tau]", "\[Element]", "Reals"]], ",", RowBox[List["0", "<", "\[Tau]", "<", "1"]]]], "}"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", RowBox[List["1", "-", RowBox[List[SuperscriptBox["\[Tau]", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["1", "-", RowBox[List[SuperscriptBox["\[Tau]", "2"], " ", SuperscriptBox["z", "2"]]]]], "<", "0"]], "&&", RowBox[List[RowBox[List["Im", "[", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox["\[Tau]", "2"], " ", SuperscriptBox["z", "2"]]]]], "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox["\[Tau]", "2"], " ", SuperscriptBox["z", "2"]]]]], "<", "0"]]]], ")"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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