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; }

 KelvinBer

 http://functions.wolfram.com/03.18.20.0016.01

 Input Form

 D[KelvinBer[\[Nu], z], {z, n}] == (2^(-1 + n - 2 \[Nu]) Sqrt[Pi] z^(-n + \[Nu]) Gamma[1 + \[Nu]] (E^((3 I Pi \[Nu])/2) HypergeometricPFQRegularized[ {(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1/2) (1 - n + \[Nu]), (1/2) (2 - n + \[Nu]), 1 + \[Nu]}, (I z^2)/4] + HypergeometricPFQRegularized[{(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1/2) (1 - n + \[Nu]), (1/2) (2 - n + \[Nu]), 1 + \[Nu]}, -((I z^2)/4)]))/E^((3/4) I Pi \[Nu]) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "+", "\[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["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]], "]"]]]], "+", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n ber ν ( z ) z n 2 n - 2 ν - 1 1 4 ( - 3 ) π ν π z ν - n Γ ( ν + 1 ) ( 3 π ν 2 2 F ~ 3 ( ν + 1 2 , ν + 2 2 ; 1 2 ( - n + ν + 1 ) , 1 2 ( - n + ν + 2 ) , ν + 1 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[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[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]", "+", "1"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]", "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + 2 F ~ 3 ( ν + 1 2 , ν + 2 2 ; 1 2 ( - n + ν + 1 ) , 1 2 ( - n + ν + 2 ) , ν + 1 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[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[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]", "+", "1"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]", "+", "2"]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], " "]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] ) /; n Condition z n KelvinBer ν z 2 n -1 2 ν -1 1 4 -3 ν 1 2 z ν -1 n Gamma ν 1 3 ν 2 -1 HypergeometricPFQRegularized ν 1 2 -1 ν 2 2 -1 1 2 -1 n ν 1 1 2 -1 n ν 2 ν 1 z 2 4 -1 HypergeometricPFQRegularized ν 1 2 -1 ν 2 2 -1 1 2 -1 n ν 1 1 2 -1 n ν 2 ν 1 -1 z 2 4 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n", "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "3"]], ")"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "+", "\[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["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]], "]"]]]], "+", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], ")"]]]]]], "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02