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http://functions.wolfram.com/03.19.06.0021.01
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KelvinKei[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /;
Subscript[F, n][z, \[Nu]] == (-((z^\[Nu] Gamma[-\[Nu]])/2^(\[Nu] + 1)))
Sum[(Sin[(1/4) Pi (\[Nu] - 2 k)]/(Pochhammer[\[Nu] + 1, k] k!))
(z/2)^(2 k), {k, 0, n}] - (Gamma[\[Nu]]/(z^\[Nu] 2^(-\[Nu] + 1)))
Sum[(Sin[(1/4) Pi (3 \[Nu] - 2 k)]/(Pochhammer[1 - \[Nu], k] k!))
(z/2)^(2 k), {k, 0, n}] == KelvinKei[\[Nu], z] -
((-I)^n 2^(-4 - 2 n + \[Nu]) E^((3 I Pi \[Nu])/4) Pi z^(2 + 2 n - \[Nu])
Csc[Pi \[Nu]] HypergeometricPFQRegularized[{1}, {2 + n, 2 + n - \[Nu]},
-((I z^2)/4)] + (I^n 2^(-4 - 2 n + \[Nu]) Pi z^(2 + 2 n - \[Nu])
Csc[Pi \[Nu]] HypergeometricPFQRegularized[{1},
{2 + n, 2 + n - \[Nu]}, (I z^2)/4])/E^((3/4) I Pi \[Nu]) -
(-I)^n 2^(-4 - 2 n - \[Nu]) E^((I Pi \[Nu])/4) Pi z^(2 + 2 n + \[Nu])
Csc[Pi \[Nu]] HypergeometricPFQRegularized[{1}, {2 + n, 2 + n + \[Nu]},
-((I z^2)/4)] - (I^n 2^(-4 - 2 n - \[Nu]) Pi z^(2 + 2 n + \[Nu])
Csc[Pi \[Nu]] HypergeometricPFQRegularized[{1},
{2 + n, 2 + n + \[Nu]}, (I z^2)/4])/E^((1/4) I Pi \[Nu])) &&
Element[n, Integers] && n >= 0
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Date Added to functions.wolfram.com (modification date)
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