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http://functions.wolfram.com/03.19.06.0030.01
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KelvinKei[\[Nu], z] == 4^(-3 - Abs[\[Nu]]) Pi^2 z^(2 + Abs[\[Nu]])
Sin[(1/4) Pi (2 \[Nu] + Abs[\[Nu]])] HypergeometricPFQRegularized[{},
{3/2, (1/2) (2 + Abs[\[Nu]]), (1/2) (3 + Abs[\[Nu]])}, -(z^4/256)] -
4^(-1 - Abs[\[Nu]]) Pi^2 z^Abs[\[Nu]] Cos[(1/4) Pi (2 \[Nu] + Abs[\[Nu]])]
HypergeometricPFQRegularized[{}, {1/2, (1/2) (1 + Abs[\[Nu]]),
(1/2) (2 + Abs[\[Nu]])}, -(z^4/256)] +
((1/4) I Sum[(1/k!) (E^((1/4) I Pi (2 \[Nu] + Abs[\[Nu]])) -
(-1)^k/E^((1/4) (I Pi (2 \[Nu] + Abs[\[Nu]])))) (Abs[\[Nu]] - k - 1)!
((I z^2)/4)^k, {k, 0, Abs[\[Nu]] - 1}])/(z/2)^Abs[\[Nu]] +
(1/4) I ((I z)/2)^Abs[\[Nu]] E^((I Pi \[Nu])/2)
Sum[(1/(k! (k + Abs[\[Nu]])!)) (E^((-(1/4)) (I Pi Abs[\[Nu]])) -
(-1)^k E^((1/4) I Pi Abs[\[Nu]])) (2 Log[z/2] - PolyGamma[1 + k] -
PolyGamma[1 + k + Abs[\[Nu]]]) ((I z^2)/4)^k, {k, 0, Infinity}] /;
Element[\[Nu], Integers]
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TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> ν </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> ν </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> ν </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> KelvinKei </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <abs /> <ci> ν </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list /> <list> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 256 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> ν </ci> </apply> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list /> <list> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 256 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <abs /> <ci> ν </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <imaginaryi /> <pi /> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> </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 type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <apply> <abs /> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> ν </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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){kind=small}
