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/03.17.22.0001.01

 Input Form

 LaplaceTransform[KelvinBei[\[Nu], t], t, z] == (Pi z^(-3 - \[Nu]) Gamma[1 + \[Nu]] ((1 + \[Nu]) (2 + \[Nu]) Cos[(3 Pi \[Nu])/4] HypergeometricPFQRegularized[ {(3 + \[Nu])/4, (4 + \[Nu])/4, (5 + \[Nu])/4, (6 + \[Nu])/4}, {3/2, (2 + \[Nu])/2, (3 + \[Nu])/2}, -(1/z^4)] + 16 z^2 Sin[(3 Pi \[Nu])/4] HypergeometricPFQRegularized[ {(1 + \[Nu])/4, (2 + \[Nu])/4, (3 + \[Nu])/4, (4 + \[Nu])/4}, {1/2, (1 + \[Nu])/2, (2 + \[Nu])/2}, -(1/z^4)]))/2^(2 (2 + \[Nu])) /; Re[\[Nu]] > -1 && Re[z] > 1/Sqrt[2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["4", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["5", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["6", "+", "\[Nu]"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "4"]]]]]], "]"]]]], "+", RowBox[List["16", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["4", "+", "\[Nu]"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "4"]]]]]], "]"]]]]]], " ", ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", "1"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", FractionBox["1", SqrtBox["2"]]]]]]]]]]

 MathML Form

 t [ bei ν ( t ) ] ( z ) 2 - 2 ( ν + 2 ) π z - ν - 3 Γ ( ν + 1 ) ( 16 sin ( 3 π ν 4 ) z 2 4 F ~ 3 ( ν + 1 4 , ν + 2 4 , ν + 3 4 , ν + 4 4 ; 1 2 , ν + 1 2 , ν + 2 2 ; - 1 z 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "3"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "4"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "4"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + ( ν + 1 ) ( ν + 2 ) cos ( 3 π ν 4 ) 4 F ~ 3 ( ν + 3 4 , ν + 4 4 , ν + 5 4 , ν + 6 4 ; 3 2 , ν + 2 2 , ν + 3 2 ; - 1 z 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "4"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "3"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "4"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "5"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "6"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "4"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] ) /; Re ( ν ) > - 1 Re ( z ) > 1 2 Condition LaplaceTransform KelvinBei ν t t z 2 -2 ν 2 z -1 ν -3 Gamma ν 1 16 sin 3 ν 4 -1 z 2 HypergeometricPFQRegularized ν 1 4 -1 ν 2 4 -1 ν 3 4 -1 ν 4 4 -1 1 2 ν 1 2 -1 ν 2 2 -1 -1 1 z 4 -1 ν 1 ν 2 3 ν 4 -1 HypergeometricPFQRegularized ν 3 4 -1 ν 4 4 -1 ν 5 4 -1 ν 6 4 -1 3 2 ν 2 2 -1 ν 3 2 -1 -1 1 z 4 -1 ν -1 z 1 2 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]_", ",", "t_"]], "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["4", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["5", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["6", "+", "\[Nu]"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "4"]]]]]], "]"]]]], "+", RowBox[List["16", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["4", "+", "\[Nu]"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "4"]]]]]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", "1"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", FractionBox["1", SqrtBox["2"]]]]]]]]]]]]

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

 2007-05-02