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 KelvinBei

 http://functions.wolfram.com/03.17.26.0051.01

 Input Form

 Hypergeometric0F1[1 - \[Nu], (I z^2)/4] KelvinBei[\[Nu], z] == ((-2^(-(1/2) - \[Nu])) I Sqrt[Pi] z^\[Nu] Gamma[1 - \[Nu]] (E^((3 I Pi \[Nu])/2) MeijerG[{{(1 + \[Nu])/2}, {(1 + 2 \[Nu])/4}}, {{\[Nu]/2}, {-(\[Nu]/2), (3 \[Nu])/2, (1 + 2 \[Nu])/4}}, I z^2] - 2^((3 \[Nu] - 1)/2) MeijerG[{{}, {(2 - \[Nu])/4}}, {{\[Nu]/4, (\[Nu] + 2)/4}, {(3 \[Nu])/4, -(\[Nu]/4), (2 - \[Nu])/4}}, -(z^4/64)]))/(E^((3/4) I Pi \[Nu]) ((-1)^(1/4) z)^\[Nu]) /; Inequality[-(Pi/2), Less, Arg[z], LessEqual, 0]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]], "]"]], RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], "4"]]], "}"]]]], "}"]], ",", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "-", "1"]], "2"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Nu]", "4"], ",", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"], ",", RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]]]]]

 MathML Form

 0 F 1 ( ; 1 - ν ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "\[Nu]"]], Hypergeometric0F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], Hypergeometric0F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric0F1] bei ν ( z ) - 2 - ν - 1 2 1 4 ( - 3 ) π ν π z ν ( - 1 4 z ) - ν Γ ( 1 - ν ) ( 3 π ν 2 G 2 , 4 1 , 1 ( z 2 ν + 1 2 , 1 4 ( 2 ν + 1 ) ν 2 , - ν 2 , 3 ν 2 , 1 4 ( 2 ν + 1 ) ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["1", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - 2 1 2 ( 3 ν - 1 ) G 1 , 5 2 , 0 ( - z 4 64 2 - ν 4 ν 4 , ν + 2 4 , 3 ν 4 , - ν 4 , 2 - ν 4 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "5"]], RowBox[List["2", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]], List[RowBox[List[TagBox[FractionBox["\[Nu]", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] ) /; - π 2 < arg ( z ) 0 Condition Hypergeometric0F1 1 -1 ν z 2 4 -1 KelvinBei ν z -1 2 -1 ν -1 1 2 1 4 -3 ν 1 2 z ν -1 1 4 z -1 ν Gamma 1 -1 ν 3 ν 2 -1 MeijerG ν 1 2 -1 1 4 2 ν 1 ν 2 -1 -1 ν 2 -1 3 ν 2 -1 1 4 2 ν 1 z 2 -1 2 1 2 3 ν -1 MeijerG 2 -1 ν 4 -1 ν 4 -1 ν 2 4 -1 3 ν 4 -1 -1 ν 4 -1 2 -1 ν 4 -1 -1 z 4 64 -1 Inequality -1 2 -1 z 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]_"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z_", "2"]]], "4"]]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "3"]], ")"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["\[Nu]", "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "-", "1"]], ")"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["\[Nu]", "4"], ",", FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"], ",", RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"]]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "64"]]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]]]]]]]

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

 2007-05-02