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/03.13.06.0028.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "8"]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "4"], ",", FractionBox["3", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", 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"4"]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "8"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["5", "4"]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["-", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "8"]]], "+", 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 MathML Form

 bei ( z ) - 1 2 2 π - z ( - z 2 ( - 1 8 ( 3 π ) + z 2 4 F 1 ( 1 4 , 1 4 , 3 4 , 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[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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] + 3 π 8 - z 2 4 F 1 ( 1 4 , 1 4 , 3 4 , 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[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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 ( π 8 - z 2 4 F 1 ( 1 4 , 1 4 , 3 4 , 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[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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] - - 1 8 ( π ) + z 2 4 F 1 ( 1 4 , 1 4 , 3 4 , 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[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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 8 z ( z 2 ( 3 π 8 - z 2 4 F 1 ( 3 4 , 3 4 , 5 4 , 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[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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] - - 1 8 ( 3 π ) + z 2 4 F 1 ( 3 4 , 3 4 , 5 4 , 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[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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] ) + - z 2 ( - - 1 8 ( π ) + z 2 4 F 1 ( 3 4 , 3 4 , 5 4 , 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[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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] - π 8 - z 2 4 F 1 ( 3 4 , 3 4 , 5 4 , 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[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[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 z -1 1 2 2 1 2 -1 z 1 2 -1 -1 z 2 1 2 -1 -1 1 8 3 z 2 1 2 -1 HypergeometricPFQ 1 4 1 4 3 4 3 4 1 2 z 2 -1 3 8 -1 -1 z 2 1 2 -1 HypergeometricPFQ 1 4 1 4 3 4 3 4 1 2 -1 z 2 -1 z 2 1 2 -1 8 -1 -1 z 2 1 2 -1 HypergeometricPFQ 1 4 1 4 3 4 3 4 1 2 z 2 -1 -1 -1 1 8 z 2 1 2 -1 HypergeometricPFQ 1 4 1 4 3 4 3 4 1 2 -1 z 2 -1 1 8 z -1 z 2 1 2 -1 3 8 -1 -1 z 2 1 2 -1 HypergeometricPFQ 3 4 3 4 5 4 5 4 3 2 z 2 -1 -1 -1 1 8 3 z 2 1 2 -1 HypergeometricPFQ 3 4 3 4 5 4 5 4 3 2 -1 z 2 -1 -1 z 2 1 2 -1 -1 -1 1 8 z 2 1 2 -1 HypergeometricPFQ 3 4 3 4 5 4 5 4 3 2 z 2 -1 -1 8 -1 -1 z 2 1 2 -1 HypergeometricPFQ 3 4 3 4 5 4 5 4 3 2 -1 z 2 -1 Inequality 2 -1 z Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBei", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "8"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "4"], ",", FractionBox["3", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02