![](/common/images/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
http://functions.wolfram.com/09.42.20.0005.01
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
D[InverseJacobiDS[z, m], m] ==
(1/2) ((z (2 m - 1 + z^2))/((m - 1) m Sqrt[m - 1 + z^2] Sqrt[m + z^2]) +
(I EllipticE[ArcSin[z/Sqrt[1 - m]], (m - 1)/m])/Sqrt[m] +
(I EllipticE[I ArcSinh[z/Sqrt[m]], m/(m - 1)])/Sqrt[m - 1] +
((m - 1) EllipticE[1/(1 - m)] - m EllipticK[1/(1 - m)])/(Sqrt[m - 1] m) +
((m - 1) EllipticK[1/m] - m EllipticE[1/m])/((m - 1) Sqrt[m])) /;
Element[z, Reals] && m > 1
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "m"], RowBox[List["InverseJacobiDS", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "1", "+", SuperscriptBox["z", "2"]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", "m", " ", SqrtBox[RowBox[List["m", "-", "1", "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List["m", "+", SuperscriptBox["z", "2"]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", FractionBox["z", SqrtBox[RowBox[List["1", "-", "m"]]]], "]"]], ",", FractionBox[RowBox[List["m", "-", "1"]], "m"]]], "]"]]]], SqrtBox["m"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox["z", SqrtBox["m"]], "]"]]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]]]], SqrtBox[RowBox[List["m", "-", "1"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["EllipticE", "[", FractionBox["1", RowBox[List["1", "-", "m"]]], "]"]]]], "-", RowBox[List["m", " ", RowBox[List["EllipticK", "[", FractionBox["1", RowBox[List["1", "-", "m"]]], "]"]]]]]], RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", "m"]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]], "-", RowBox[List["m", " ", RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SqrtBox["m"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", ">", "1"]]]]]]]]
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mo> ∂ </mo> <mrow> <msup> <mi> ds </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <mi> m </mi> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> m </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> m </mi> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> m </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mtext> </mtext> </mrow> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ❘ </mo> <mfrac> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mtext> </mtext> </mrow> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <msqrt> <mi> m </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ❘ </mo> <mfrac> <mi> m </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> m </ci> </bvar> <apply> <ci> InverseJacobiDS </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <ci> m </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> m </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticE </ci> <apply> <arcsin /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <ci> m </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> m </ci> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> z </ci> <reals /> </apply> <apply> <gt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/clear.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| | ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["m_"]]], RowBox[List["InverseJacobiDS", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m"]], "-", "1", "+", SuperscriptBox["z", "2"]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", "m", " ", SqrtBox[RowBox[List["m", "-", "1", "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List["m", "+", SuperscriptBox["z", "2"]]]]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["ArcSin", "[", FractionBox["z", SqrtBox[RowBox[List["1", "-", "m"]]]], "]"]], ",", FractionBox[RowBox[List["m", "-", "1"]], "m"]]], "]"]]]], SqrtBox["m"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["EllipticE", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSinh", "[", FractionBox["z", SqrtBox["m"]], "]"]]]], ",", FractionBox["m", RowBox[List["m", "-", "1"]]]]], "]"]]]], SqrtBox[RowBox[List["m", "-", "1"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["EllipticE", "[", FractionBox["1", RowBox[List["1", "-", "m"]]], "]"]]]], "-", RowBox[List["m", " ", RowBox[List["EllipticK", "[", FractionBox["1", RowBox[List["1", "-", "m"]]], "]"]]]]]], RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", "m"]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]]]], "-", RowBox[List["m", " ", RowBox[List["EllipticE", "[", FractionBox["1", "m"], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SqrtBox["m"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["z", "\[Element]", "Reals"]], "&&", RowBox[List["m", ">", "1"]]]]]]]]]] |
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
| ![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
Date Added to functions.wolfram.com (modification date)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
![](/images/home/spacer.gif)
|
|
![](/common/images/spacer.gif) |
|
|