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 KelvinKer

 http://functions.wolfram.com/03.20.06.0047.01

 Input Form

 KelvinKer[\[Nu], z] \[Proportional] (1/64) E^(-(((1 + I) z)/Sqrt[2]) - (5 I Pi \[Nu])/4) Sqrt[Pi/2] z^(-2 - \[Nu]) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) (-2 I E^((1 + I) Sqrt[2] z + (5 I Pi \[Nu])/2) z^(1 + 2 \[Nu]) + 2 I E^((1 + I) Sqrt[2] z + I Pi \[Nu]) z ((-1)^(3/4) z)^(2 \[Nu]) + ((-1)^(3/4) z)^(2 \[Nu]) (z - (-1)^(3/4) Sqrt[I z^2]) + E^((7 I Pi \[Nu])/2) z^(2 \[Nu]) (-z + (-1)^(3/4) Sqrt[I z^2]) + E^((3 I Pi \[Nu])/2) z^(2 \[Nu]) (z + (-1)^(3/4) Sqrt[I z^2]) - E^(2 I Pi \[Nu]) ((-1)^(3/4) z)^(2 \[Nu]) (z + (-1)^(3/4) Sqrt[I z^2])) (-1 + 4 \[Nu]^2) (1 + I Cot[Pi \[Nu]]) HypergeometricPFQ[ {3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 3/4 + \[Nu]/2, 5/4 + \[Nu]/2}, {3/2}, -(I/z^2)] - (1/64) E^(-(((1 + I) z)/Sqrt[2]) + (I Pi \[Nu])/4) Sqrt[Pi/2] z^(-2 - \[Nu]) ((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) (2 E^(I Sqrt[2] z + (3 I Pi \[Nu])/2) z^(1 + 2 \[Nu]) - 2 E^(I (Sqrt[2] z + Pi \[Nu])) z ((-(-1)^(1/4)) z)^(2 \[Nu]) + E^(Sqrt[2] z) ((-(-1)^(1/4)) z)^(2 \[Nu]) (I z - (-1)^(3/4) Sqrt[(-I) z^2]) + E^(Sqrt[2] z + (5 I Pi \[Nu])/2) z^(2 \[Nu]) ((-I) z + (-1)^(3/4) Sqrt[(-I) z^2]) + E^(Sqrt[2] z + (I Pi \[Nu])/2) z^(2 \[Nu]) (I z + (-1)^(3/4) Sqrt[(-I) z^2]) - E^(Sqrt[2] z + 2 I Pi \[Nu]) ((-(-1)^(1/4)) z)^(2 \[Nu]) (I z + (-1)^(3/4) Sqrt[(-I) z^2])) (-1 + 4 \[Nu]^2) (1 + I Cot[Pi \[Nu]]) HypergeometricPFQ[ {3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 3/4 + \[Nu]/2, 5/4 + \[Nu]/2}, {3/2}, I/z^2] - (1/4) E^(-(((1 + I) z)/Sqrt[2]) - (I Pi \[Nu])/4) Sqrt[Pi/2] z^(-1 - \[Nu]) ((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) (((-1)^(1/4) E^(I Sqrt[2] z) z^(1 + 2 \[Nu]) - E^(Sqrt[2] z + (I Pi \[Nu])/2) ((-(-1)^(1/4)) z)^(2 \[Nu]) Sqrt[(-I) z^2]) Cot[Pi \[Nu]] + E^(Sqrt[2] z) z^(2 \[Nu]) Cos[Pi \[Nu]] (I ((-1)^(3/4) z + Sqrt[(-I) z^2]) + Sqrt[(-I) z^2] Cot[Pi \[Nu]]) + (-1)^(3/4) z (E^(I Sqrt[2] z) z^(2 \[Nu]) + I E^(Sqrt[2] z + (I Pi \[Nu])/2) ((-(-1)^(1/4)) z)^(2 \[Nu]) + I E^((1/2) I (2 Sqrt[2] z + Pi \[Nu])) ((-(-1)^(1/4)) z)^(2 \[Nu]) Csc[Pi \[Nu]] - E^(Sqrt[2] z) z^(2 \[Nu]) Sin[Pi \[Nu]])) HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/4 + \[Nu]/2, 3/4 + \[Nu]/2}, {1/2}, I/z^2] - ((1/4) E^(-(((1 + I) z)/Sqrt[2]) - (5 I Pi \[Nu])/4) Sqrt[Pi/2] ((-1)^(3/4) z)^(-(3/2) - \[Nu]) (((-E^((1 + I) Sqrt[2] z + (5 I Pi \[Nu])/2)) z^(1 + 2 \[Nu]) + (-1)^(1/4) ((-1)^(3/4) z)^(2 \[Nu]) Sqrt[I z^2]) Cot[Pi \[Nu]] + E^((5 I Pi \[Nu])/2) z^(2 \[Nu]) Cos[Pi \[Nu]] (z + (-1)^(3/4) Sqrt[I z^2] - (-1)^(1/4) Sqrt[I z^2] Cot[Pi \[Nu]]) + z (I E^((1 + I) Sqrt[2] z + (5 I Pi \[Nu])/2) z^(2 \[Nu]) + ((-1)^(3/4) z)^(2 \[Nu]) + E^((1 + I) Sqrt[2] z) ((-1)^(3/4) z)^(2 \[Nu]) Csc[Pi \[Nu]] - I E^((5 I Pi \[Nu])/2) z^(2 \[Nu]) Sin[Pi \[Nu]])) HypergeometricPFQ[ {1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/4 + \[Nu]/2, 3/4 + \[Nu]/2}, {1/2}, -(I/z^2)])/z^\[Nu] /; (Abs[z] -> Infinity) && !Element[\[Nu], Integers]

 Standard Form

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 MathML Form

 ker ν ( z ) - 1 64 π ν 4 - ( 1 + ) z 2 π 2 ( - - 1 4 z ) - ν - 1 2 z - ν - 2 ( 2 z + 5 π ν 2 ( ( - 1 ) 3 / 4 - z 2 - z ) z 2 ν + 2 z + π ν 2 ( z + ( - 1 ) 3 / 4 - z 2 ) z 2 ν + 2 2 z + 3 π ν 2 z 2 ν + 1 - 2 ( 2 z + π ν ) ( - - 1 4 z ) 2 ν z - 2 z + 2 π ν ( - - 1 4 z ) 2 ν ( z + ( - 1 ) 3 / 4 - z 2 ) + 2 z ( - - 1 4 z ) 2 ν ( z - ( - 1 ) 3 / 4 - z 2 ) ) ( 4 ν 2 - 1 ) ( cot ( π ν ) + 1 ) 4 F 1 ( 3 4 - ν 2 , 5 4 - ν 2 , ν 2 + 3 4 , ν 2 + 5 4 ; 3 2 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] + 1 64 π 2 - ( 1 + ) z 2 - 5 π ν 4 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 z - ν - 2 ( 7 π ν 2 ( ( - 1 ) 3 / 4 z 2 - z ) z 2 ν + 3 π ν 2 ( z + ( - 1 ) 3 / 4 z 2 ) z 2 ν - 2 2 ( 1 + ) z + 5 π ν 2 z 2 ν + 1 + 2 2 ( 1 + ) z + π ν ( ( - 1 ) 3 / 4 z ) 2 ν z - 2 π ν ( ( - 1 ) 3 / 4 z ) 2 ν ( z + ( - 1 ) 3 / 4 z 2 ) + ( ( - 1 ) 3 / 4 z ) 2 ν ( z - ( - 1 ) 3 / 4 z 2 ) ) ( 4 ν 2 - 1 ) ( cot ( π ν ) + 1 ) 4 F 1 ( 3 4 - ν 2 , 5 4 - ν 2 , ν 2 + 3 4 , ν 2 + 5 4 ; 3 2 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] - 1 4 - ( 1 + ) z 2 - π ν 4 π 2 ( - - 1 4 z ) - ν - 1 2 z - ν - 1 ( 2 z cos ( π ν ) ( ( ( - 1 ) 3 / 4 z + - z 2 ) + - z 2 cot ( π ν ) ) z 2 ν + ( - 1 ) 3 / 4 ( 2 z z 2 ν - 2 z sin ( π ν ) z 2 ν + 2 z + π ν 2 ( - - 1 4 z ) 2 ν + 1 2 ( 2 2 z + π ν ) ( - - 1 4 z ) 2 ν csc ( π ν ) ) z + ( - 1 4 2 z z 2 ν + 1 - 2 z + π ν 2 ( - - 1 4 z ) 2 ν - z 2 ) cot ( π ν ) ) 4 F 1 ( 1 4 - ν 2 , 3 4 - ν 2 , ν 2 + 1 4 , ν 2 + 3 4 ; 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( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Proportional KelvinKer ν z -1 1 64 ν 4 -1 -1 1 z 2 1 2 -1 2 -1 1 2 -1 -1 1 4 z -1 ν -1 1 2 z -1 ν -2 2 1 2 z 5 ν 2 -1 -1 3 4 -1 z 2 1 2 -1 z z 2 ν 2 1 2 z ν 2 -1 z -1 3 4 -1 z 2 1 2 z 2 ν 2 2 1 2 z 3 ν 2 -1 z 2 ν 1 -1 2 2 1 2 z ν -1 -1 1 4 z 2 ν z -1 2 1 2 z 2 ν -1 -1 1 4 z 2 ν z -1 3 4 -1 z 2 1 2 2 1 2 z -1 -1 1 4 z 2 ν z -1 -1 3 4 -1 z 2 1 2 4 ν 2 -1 ν 1 HypergeometricPFQ 3 4 -1 ν 2 -1 5 4 -1 ν 2 -1 ν 2 -1 3 4 ν 2 -1 5 4 3 2 z 2 -1 1 64 2 -1 1 2 -1 1 z 2 1 2 -1 -1 5 ν 4 -1 -1 3 4 z -1 ν -1 1 2 z -1 ν -2 7 ν 2 -1 -1 3 4 z 2 1 2 -1 z z 2 ν 3 ν 2 -1 z -1 3 4 z 2 1 2 z 2 ν -1 2 2 1 2 1 z 5 ν 2 -1 z 2 ν 1 2 2 1 2 1 z ν -1 3 4 z 2 ν z -1 2 ν -1 3 4 z 2 ν z -1 3 4 z 2 1 2 -1 3 4 z 2 ν z -1 -1 3 4 z 2 1 2 4 ν 2 -1 ν 1 HypergeometricPFQ 3 4 -1 ν 2 -1 5 4 -1 ν 2 -1 ν 2 -1 3 4 ν 2 -1 5 4 3 2 -1 z 2 -1 -1 1 4 -1 1 z 2 1 2 -1 -1 ν 4 -1 2 -1 1 2 -1 -1 1 4 z -1 ν -1 1 2 z -1 ν -1 2 1 2 z ν -1 3 4 z -1 z 2 1 2 -1 z 2 1 2 ν z 2 ν -1 3 4 2 1 2 z z 2 ν -1 2 1 2 z ν z 2 ν 2 1 2 z ν 2 -1 -1 -1 1 4 z 2 ν 1 2 2 2 1 2 z ν -1 -1 1 4 z 2 ν ν z -1 1 4 2 1 2 z z 2 ν 1 -1 2 1 2 z ν 2 -1 -1 -1 1 4 z 2 ν -1 z 2 1 2 ν HypergeometricPFQ 1 4 -1 ν 2 -1 3 4 -1 ν 2 -1 ν 2 -1 1 4 ν 2 -1 3 4 1 2 z 2 -1 -1 1 4 -1 1 z 2 1 2 -1 -1 5 ν 4 -1 2 -1 1 2 -1 3 4 z -1 ν -1 3 2 z -1 ν 5 ν 2 -1 ν z -1 -1 1 4 z 2 1 2 ν -1 3 4 z 2 1 2 z 2 ν 2 1 2 1 z 5 ν 2 -1 z 2 ν -1 5 ν 2 -1 ν z 2 ν -1 3 4 z 2 ν 1 2 1 2 z -1 3 4 z 2 ν ν z -1 1 4 -1 3 4 z 2 ν z 2 1 2 -1 2 1 2 1 z 5 ν 2 -1 z 2 ν 1 ν HypergeometricPFQ 1 4 -1 ν 2 -1 3 4 -1 ν 2 -1 ν 2 -1 1 4 ν 2 -1 3 4 1 2 -1 z 2 -1 Rule z ν [/itex]

 Rule Form

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

 2007-05-02