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 KelvinKei

 http://functions.wolfram.com/03.19.06.0030.01

 Input Form

 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]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", RowBox[List["Abs", "[", "\[Nu]", "]"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ")"]]]]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"], ")"]], RowBox[List["Abs", "[", "\[Nu]", "]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]

 MathML Form

 kei ν ( z ) 4 ( z 2 ) - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 ( 1 4 π ( 2 ν + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) - ( - 1 ) k - 1 4 ( π ( 2 ν + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ) ) ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 1 ) ! k ! ( z 2 4 ) k - 4 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 π 2 z "\[LeftBracketingBar]" ν "\[RightBracketingBar]" cos ( 1 4 π ( 2 ν + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ) 0 F ~ 3 ( ; 1 2 , 1 2 ( ν + 1 ) , 1 2 ( ν + 2 ) ; - z 4 256 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]], "+", "1"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]], "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + 4 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 3 π 2 z "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 2 sin ( 1 4 π ( 2 ν + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ) 0 F ~ 3 ( ; 3 2 , 1 2 ( ν + 2 ) , 1 2 ( ν + 3 ) ; - z 4 256 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]], "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]], "+", "3"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + 4 ( z 2 ) "\[LeftBracketingBar]" ν "\[RightBracketingBar]" π ν 2 k = 0 ( - 1 4 ( π "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) - ( - 1 ) k 1 4 π "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ( 2 log ( z 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ) k ! ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ! ( z 2 4 ) k /; ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition KelvinKei ν z 4 -1 z 2 -1 -1 ν k 0 ν -1 1 4 2 ν ν -1 -1 k -1 1 4 2 ν ν ν -1 k -1 k -1 z 2 4 -1 k -1 4 -1 ν -1 2 z ν 1 4 2 ν ν HypergeometricPFQRegularized 1 2 1 2 ν 1 1 2 ν 2 -1 z 4 256 -1 4 -1 ν -3 2 z ν 2 1 4 2 ν ν HypergeometricPFQRegularized 3 2 1 2 ν 2 1 2 ν 3 -1 z 4 256 -1 4 -1 z 2 -1 ν ν 2 -1 k 0 -1 1 4 ν -1 -1 k 1 4 ν 2 z 2 -1 -1 PolyGamma k 1 -1 PolyGamma k ν 1 k k ν -1 z 2 4 -1 k ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", RowBox[List["Abs", "[", "\[Nu]", "]"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ")"]]]]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"], ")"]], RowBox[List["Abs", "[", "\[Nu]", "]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]]]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02