html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 KelvinBei

 http://functions.wolfram.com/03.17.26.0048.01

 Input Form

 Hypergeometric0F1[1 + \[Nu], (I z^2)/4] KelvinBei[\[Nu], z] == (-((I Sqrt[Pi])/(2 Sqrt[2]))) Gamma[1 + \[Nu]] (E^((3 I Pi \[Nu])/4) MeijerG[{{(3 - \[Nu])/4, (1 - \[Nu])/4}, {}}, {{\[Nu]/4}, {-(\[Nu]/4), (2 + \[Nu])/4, (2 - \[Nu])/4, (2 - 3 \[Nu])/4, -((3 \[Nu])/4)}}, z^4/16] + I E^((3 I Pi \[Nu])/4) MeijerG[{{(3 - \[Nu])/4, (1 - \[Nu])/4}, {}}, {{(2 + \[Nu])/4}, {-(\[Nu]/4), \[Nu]/4, (2 - \[Nu])/4, (2 - 3 \[Nu])/4, -((3 \[Nu])/4)}}, z^4/16] - (Pi 2^((1 - \[Nu])/2) MeijerG[{{}, {(2 + \[Nu])/4}}, {{\[Nu]/4}, {-(\[Nu]/4), (2 - \[Nu])/4, -((3 \[Nu])/4), (2 + \[Nu])/4}}, z^4/64])/E^((3 I Pi \[Nu])/4)) /; -(Pi/4) <= Arg[z] <= Pi/4

 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["-", FractionBox[RowBox[List["\[ImaginaryI]", SqrtBox["\[Pi]"]]], RowBox[List["2", SqrtBox["2"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["\[Nu]", "4"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], "4"], ",", RowBox[List["-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "16"]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox["\[Nu]", "4"], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], "4"], ",", RowBox[List["-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "16"]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["3", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]]]], SuperscriptBox["2", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "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["{", FractionBox["\[Nu]", "4"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], ",", RowBox[List["-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "4"]]], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "64"]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "4"]]], "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "4"]]]]]]]

 MathML Form

 0 F 1 ( ; ν + 1 ; 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["\[Nu]", "+", "1"]], 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 ) - π 2 2 Γ ( ν + 1 ) ( - π - 1 4 ( 3 π ν ) 2 1 - ν 2 G 1 , 5 1 , 0 ( z 4 64 ν + 2 4 ν 4 , - ν 4 , 2 - ν 4 , - 1 4 ( 3 ν ) , ν + 2 4 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "5"]], RowBox[List["1", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "4"], "64"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]], List[RowBox[List[TagBox[FractionBox["\[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]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Nu]"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] + 3 π ν 4 G 2 , 6 1 , 2 ( z 4 16 3 - ν 4 , 1 - ν 4 ν 4 , - ν 4 , ν + 2 4 , 2 - ν 4 , 1 4 ( 2 - 3 ν ) , - 1 4 ( 3 ν ) ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "6"]], RowBox[List["1", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "4"], "16"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["3", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[FractionBox["\[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["\[Nu]", "+", "2"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Nu]"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] + 3 π ν 4 G 2 , 6 1 , 2 ( z 4 16 3 - ν 4 , 1 - ν 4 ν + 2 4 , - ν 4 , ν 4 , 2 - ν 4 , 1 4 ( 2 - 3 ν ) , - 1 4 ( 3 ν ) ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "6"]], RowBox[List["1", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "4"], "16"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["3", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["\[Nu]", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Nu]"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] ) /; - π 4 arg ( z ) π 4 Condition Hypergeometric0F1 ν 1 z 2 4 -1 KelvinBei ν z -1 1 2 2 2 1 2 -1 Gamma ν 1 -1 -1 1 4 3 ν 2 1 -1 ν 2 -1 MeijerG ν 2 4 -1 ν 4 -1 -1 ν 4 -1 2 -1 ν 4 -1 -1 1 4 3 ν ν 2 4 -1 z 4 64 -1 3 ν 4 -1 MeijerG 3 -1 ν 4 -1 1 -1 ν 4 -1 ν 4 -1 -1 ν 4 -1 ν 2 4 -1 2 -1 ν 4 -1 1 4 2 -1 3 ν -1 1 4 3 ν z 4 16 -1 3 ν 4 -1 MeijerG 3 -1 ν 4 -1 1 -1 ν 4 -1 ν 2 4 -1 -1 ν 4 -1 ν 4 -1 2 -1 ν 4 -1 1 4 2 -1 3 ν -1 1 4 3 ν z 4 16 -1 -1 4 -1 z 4 -1 [/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["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["\[Nu]", "4"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Nu]"]], ")"]]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "16"]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "4"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox["\[Nu]", "4"], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["3", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Nu]"]], ")"]]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "16"]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", SuperscriptBox["2", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "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["{", FractionBox["\[Nu]", "4"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox[RowBox[List["2", "-", "\[Nu]"]], "4"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Nu]"]], ")"]]]], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "4"], "64"]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["2"]]]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "4"]]], "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "4"]]]]]]]]]

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

 2007-05-02