EllipticPi[n, z, m] \[Proportional] EllipticF[z, m] + (-1)^Round[Re[z]/Pi] (n/3) (AppellF1[3/2, 1/2, 1/2, 5/2, Sin[z]^2, m Sin[z]^2] Sin[z]^3 + (3/5) n AppellF1[5/2, 1/2, 1/2, 7/2, Sin[z]^2, m Sin[z]^2] Sin[z]^5 + (3/7) n^2 AppellF1[7/2, 1/2, 1/2, 9/2, Sin[z]^2, m Sin[z]^2] Sin[z]^7 + \[Ellipsis]) + ((Pi n)/2) (Hypergeometric2F1[3/2, 1/2, 2, m] + (3/4) Hypergeometric2F1[5/2, 1/2, 3, m] n + (5/8) Hypergeometric2F1[7/2, 1/2, 4, m] n^2 + \[Ellipsis]) Round[Re[z]/Pi] /; (n -> 0)

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</ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <ms> 2 </ms> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> m </ms> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ci> Hypergeometric2F1 </ci> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 4 </ms> </apply> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#62387; </ms> <ms> 2 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> &#8289; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 2 </ms> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <ms> 3 </ms> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> m </ms> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ci> Hypergeometric2F1 </ci> </apply> <ms> n </ms> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 5 </ms> <ms> 8 </ms> </apply> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#62387; </ms> <ms> 2 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> &#8289; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 7 </ms> <ms> 2 </ms> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <ms> 4 </ms> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> m </ms> <ci> Hypergeometric2F1 </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ci> Hypergeometric2F1 </ci> </apply> <apply> <ci> SuperscriptBox </ci> <ms> n </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> + </ms> <ms> &#8230; </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#62754; </ms> <ms> 0 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

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2001-10-29