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 KelvinBei

 http://functions.wolfram.com/03.17.06.0043.01

 Input Form

 KelvinBei[\[Nu], z] \[Proportional] (-(1/(2 Sqrt[2 Pi] Sqrt[-z]))) (((E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 - (3 Pi I)/8) HypergeometricPFQ[ {1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/4 + \[Nu]/2, 3/4 + \[Nu]/2}, {1/2}, I/z^2] + E^(-((I z)/Sqrt[2]) + (6 I Pi \[Nu])/4 + (3 Pi I)/8) HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/4 + \[Nu]/2, 3/4 + \[Nu]/2}, {1/2}, -(I/z^2)])/E^(z/Sqrt[2]) + E^(z/Sqrt[2]) ((-E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 - (Pi I)/8)) HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/4 + \[Nu]/2, 3/4 + \[Nu]/2}, {1/2}, -(I/z^2)] + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 + (Pi I)/8) HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2, 1/4 + \[Nu]/2, 3/4 + \[Nu]/2}, {1/2}, I/z^2])) + ((1 - \[Nu]^2)/(8 z)) (((-E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 - (Pi I)/8)) HypergeometricPFQ[{3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 3/4 + \[Nu]/2, 5/4 + \[Nu]/2}, {3/2}, I/z^2] - E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 + (Pi I)/8) HypergeometricPFQ[{3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 3/4 + \[Nu]/2, 5/4 + \[Nu]/2}, {3/2}, -(I/z^2)])/E^(z/Sqrt[2]) + E^(z/Sqrt[2]) ((-E^((I z)/Sqrt[2] + (5 I Pi \[Nu])/2 - (3 Pi I)/8)) HypergeometricPFQ[{3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 3/4 + \[Nu]/2, 5/4 + \[Nu]/2}, {3/2}, -(I/z^2)] + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 + (3 Pi I)/8) HypergeometricPFQ[{3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 3/4 + \[Nu]/2, 5/4 + \[Nu]/2}, {3/2}, I/z^2]))) /; Inequality[Pi/2, Less, Arg[z], LessEqual, Pi] && (Abs[z] -> Infinity)

 Standard Form

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 MathML Form

 bei ν ( z ) - 1 2 2 π - z ( z 2 ( - z 2 + 3 π ν 2 + π 8 4 F 1 ( 1 4 - ν 2 , 3 4 - ν 2 , ν 2 + 1 4 , ν 2 + 3 4 ; 1 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["1", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - z 2 + π ν 2 - π 8 4 F 1 ( 1 4 - ν 2 , 3 4 - ν 2 , ν 2 + 1 4 , ν 2 + 3 4 ; 1 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["1", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) + - z 2 ( z 2 + π ν 2 - 3 π 8 4 F 1 ( 1 4 - ν 2 , 3 4 - ν 2 , ν 2 + 1 4 , ν 2 + 3 4 ; 1 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["1", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + - z 2 + 6 π ν 4 + 3 π 8 4 F 1 ( 1 4 - ν 2 , 3 4 - ν 2 , ν 2 + 1 4 , ν 2 + 3 4 ; 1 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["1", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["1", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) + 1 - ν 2 8 z ( z 2 ( - z 2 + 3 π ν 2 + 3 π 8 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]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - z 2 + 5 π ν 2 - 3 π 8 4 F 1 ( 3 4 - ν 2 , 5 4 - ν 2 , ν 2 + 3 4 , ν 2 + 5 4 ; 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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]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - - z 2 + 3 π ν 2 + π 8 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]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Nu]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Nu]", "2"], "+", FractionBox["5", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) ) ) /; π 2 < arg ( z ) π ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional KelvinBei ν z -1 1 2 2 1 2 -1 z 1 2 -1 z 2 1 2 -1 -1 z 2 1 2 -1 3 ν 2 -1 8 -1 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 z 2 1 2 -1 ν 2 -1 -1 8 -1 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 -1 z 2 1 2 -1 z 2 1 2 -1 ν 2 -1 -1 3 8 -1 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 z 2 1 2 -1 6 ν 4 -1 3 8 -1 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 1 -1 ν 2 8 z -1 z 2 1 2 -1 -1 z 2 1 2 -1 3 ν 2 -1 3 8 -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 z 2 1 2 -1 5 ν 2 -1 -1 3 8 -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 z 2 1 2 -1 -1 z 2 1 2 -1 ν 2 -1 -1 8 -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 -1 z 2 1 2 -1 3 ν 2 -1 8 -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 Inequality 2 -1 z Rule z [/itex]

 Rule Form

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

 2007-05-02