KelvinKei[n, z] \[Proportional] (-((Pi 2^(-n - 2) z^n Cos[(3 n Pi)/4])/n!)) (1 - z^4/(32 (1 + n) (2 + n)) + z^8/(6144 (1 + n) (2 + n) (3 + n) (4 + n)) + \[Ellipsis]) + ((Pi 2^(-4 - n) z^(2 + n) Sin[(3 n Pi)/4])/ (1 + n)!) (1 - z^4/(96 (2 + n) (3 + n)) + z^8/(30720 (2 + n) (3 + n) (4 + n) (5 + n)) + \[Ellipsis]) + ((I/4) Sum[(((E^((3 I Pi n)/4) - (-1)^k/E^((3 I Pi n)/4)) (n - k - 1)!)/ k!) ((I z^2)/4)^k, {k, 0, n - 1}])/(z/2)^n + 2^(-2 - n) (-1)^n z^n (((2 Sin[(n Pi)/4])/n!) (EulerGamma + 2 Log[z/2] - PolyGamma[1 + n]) - (Cos[(n Pi)/4]/(2 (1 + n)!)) (-1 + EulerGamma + 2 Log[z/2] - PolyGamma[2 + n]) z^2 - (Sin[(n Pi)/4]/(16 (2 + n)!)) (-(3/2) + EulerGamma + 2 Log[z/2] - PolyGamma[3 + n]) z^4 + \[Ellipsis]) /; (z -> 0) && Element[n, Integers] && n >= 0

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-1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <eulergamma /> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> n </ci> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn 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2007-05-02