Integrate[InverseJacobiDS[z, m], m] == 2 ((Sqrt[m - 1 + z^2] Sqrt[m + z^2])/z - z (Log[(Sqrt[m - 1 + z^2] + Sqrt[m + z^2])/(2 z)] + 1) + I Sqrt[m - 1] (EllipticE[I ArcSinh[Sqrt[m - 1]/z], m/(m - 1)] - EllipticF[I ArcSinh[Sqrt[m - 1]/z], m/(m - 1)])) /; z > 0 && m > 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> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> ds </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mi> m </mi> </mrow> </msqrt> </mrow> <mi> z </mi> </mfrac> <mo> - </mo> <mrow> <mi> z </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mi> m </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mfrac> <mi> m </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> z </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> m </ci> </bvar> <apply> <ci> InverseJacobiDS </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <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> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> z </ci> <apply> <plus /> <apply> <ln /> <apply> <times /> <apply> <plus /> <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> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <plus /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <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> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticF </ci> <apply> <times /> <imaginaryi /> <apply> <arcsinh /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <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> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29