EllipticPi[n, m] \[Proportional] (Pi/(2 Sqrt[-n])) (1 + (1 + m)/(2 n) + (3 + 2 m + 3 m^2)/(8 n^2) + O[1/n^3]) + (1/(4 n)) (4 (EllipticE[m] - EllipticK[m]) - (-8 (1 + m) EllipticE[m] + 4 (2 + m) EllipticK[m])/(3 n) + (4 ((8 + 7 m + 8 m^2) EllipticE[m] - (8 + 3 m + 4 m^2) EllipticK[m]))/ (15 n^2) + O[1/n^3])

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</mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mtext> </mtext> <mi> n </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> m </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 15 </mn> <mo> &#8290; 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2007-05-02