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 KelvinKer

 http://functions.wolfram.com/03.20.26.0004.01

 Input Form

 KelvinKer[\[Nu], z] == 2^(-1 + \[Nu]) E^((-(-1)^(1/4)) z - (I Pi \[Nu])/4) Sqrt[Pi] z^\[Nu] HypergeometricU[1/2 + \[Nu], 1 + 2 \[Nu], 2 (-1)^(1/4) z] + ((-1)^(\[Nu]/4) 2^(-1 + \[Nu]) Sqrt[Pi] z^\[Nu] HypergeometricU[1/2 + \[Nu], 1 + 2 \[Nu], 2 (-1)^(3/4) z])/ E^((-1)^(3/4) z) + 2^(-3 - \[Nu]) E^((3 I Pi \[Nu])/4) z^\[Nu] ((-I) Pi - 4 Log[z] + 4 Log[(-1)^(1/4) z]) Hypergeometric0F1Regularized[ 1 + \[Nu], (I z^2)/4] + (-1)^((5 \[Nu])/4) 2^(-3 - \[Nu]) z^\[Nu] (I Pi - 4 Log[z] + 4 Log[(-1)^(3/4) z]) Hypergeometric0F1Regularized[ 1 + \[Nu], -((I z^2)/4)] /; Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], " ", "z"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]]]], ")"]], RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]], "]"]]]], "+", " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["5", " ", "\[Nu]"]], "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], "]"]]]]]], ")"]], RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]]]], "]"]]]]]]]], "/;", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]]]]]]

 MathML Form

 ker ν ( z ) 2 ν - 1 - - 1 4 z - π ν 4 π z ν U TagBox["U", HypergeometricU] ( ν + 1 2 , 2 ν + 1 , 2 - 1 4 z ) + ( - 1 ) ν / 4 2 ν - 1 - ( - 1 ) 3 / 4 z π z ν U TagBox[RowBox[List[SuperscriptBox["z", "\[Nu]"], "U"]], HypergeometricU] ( ν + 1 2 , 2 ν + 1 , 2 ( - 1 ) 3 / 4 z ) + 2 - ν - 3 3 π ν 4 ( - π - 4 log ( z ) + 4 log ( - 1 4 z ) ) z ν 0 F ~ 1 ( ; ν + 1 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric0F1Regularized, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], Hypergeometric0F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric0F1Regularized] + ( - 1 ) 5 ν 4 2 - ν - 3 ( π - 4 log ( z ) + 4 log ( ( - 1 ) 3 / 4 z ) ) z ν 0 F ~ 1 ( ; ν + 1 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric0F1Regularized, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]], Hypergeometric0F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric0F1Regularized] /; ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition KelvinKer ν z 2 ν -1 -1 -1 1 4 z -1 ν 4 -1 1 2 z ν HypergeometricU ν 1 2 2 ν 1 2 -1 1 4 z -1 ν 4 -1 2 ν -1 -1 -1 3 4 z 1 2 HypergeometricU z ν U ν 1 2 2 ν 1 2 -1 3 4 z 2 -1 ν -3 3 ν 4 -1 -1 -1 4 z 4 -1 1 4 z z ν Hypergeometric0F1Regularized ν 1 z 2 4 -1 -1 5 ν 4 -1 2 -1 ν -3 -1 4 z 4 -1 3 4 z z ν Hypergeometric0F1Regularized ν 1 -1 z 2 4 -1 ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "/", "4"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]]]], " ", "z"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]"]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], FractionBox[RowBox[List["5", " ", "\[Nu]"]], "4"]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "-", RowBox[List["4", " ", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], ")"]]]]]], "]"]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

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

 2007-05-02