InverseJacobiND[z, m] \[Proportional] ((-(I/2)) (1 + m/4 + (9 m^2)/64 + \[Ellipsis]) Log[m] + I Log[4] + (I/4) (Log[4] - 1) m + ((3 I)/128) (6 Log[4] - 7) m^2 + \[Ellipsis]) + (Sqrt[1 - z^2]/Sqrt[z^2 - 1]) (ArcTanh[z] + (1/4) (z/(z^2 - 1) + ArcTanh[z]) m + (3/64) ((z (5 z^2 - 3))/(z^2 - 1)^2 + 3 ArcTanh[z]) m^2 + \[Ellipsis]) /; (m -> 0)

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</mi> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 128 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> z </mi> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 64 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> InverseJacobiND </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <plus /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <ci> m </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 64 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; 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</ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> m </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29