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 KelvinKei

 http://functions.wolfram.com/03.19.20.0014.01

 Input Form

 D[KelvinKei[\[Nu], z], {z, n}] == 2^(-2 + n + 2 \[Nu]) I E^((3 I Pi \[Nu])/4) Pi^(3/2) z^(-n - \[Nu]) Csc[Pi \[Nu]] Gamma[1 - \[Nu]] HypergeometricPFQRegularized[{(1 - \[Nu])/2, 1 - \[Nu]/2}, {(1/2) (1 - n - \[Nu]), (1/2) (2 - n - \[Nu]), 1 - \[Nu]}, -((I z^2)/4)] - (2^(-2 + n + 2 \[Nu]) I Pi^(3/2) z^(-n - \[Nu]) Csc[Pi \[Nu]] Gamma[1 - \[Nu]] HypergeometricPFQRegularized[ {(1 - \[Nu])/2, 1 - \[Nu]/2}, {(1/2) (1 - n - \[Nu]), (1/2) (2 - n - \[Nu]), 1 - \[Nu]}, (I z^2)/4])/E^((3/4) I Pi \[Nu]) - (2^(-2 + n - 2 \[Nu]) I Pi^(3/2) z^(-n + \[Nu]) (I + Cot[Pi \[Nu]]) Gamma[1 + \[Nu]] 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]) + 2^(-2 + n - 2 \[Nu]) I E^((3 I Pi \[Nu])/4) Pi^(3/2) z^(-n + \[Nu]) (-I + Cot[Pi \[Nu]]) Gamma[1 + \[Nu]] HypergeometricPFQRegularized[{(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1/2) (1 - n + \[Nu]), (1/2) (2 - n + \[Nu]), 1 + \[Nu]}, (I z^2)/4] /; !Element[\[Nu], Integers] && 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["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", " ", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", "n", "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", 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 MathML Form

 n kei ν ( z ) z n - 2 n + 2 ν - 2 1 4 ( - 3 ) π ν π 3 / 2 csc ( π ν ) Γ ( 1 - ν ) z - n - ν 2 F ~ 3 ( 1 - ν 2 , 1 - ν 2 ; 1 2 ( - n - ν + 1 ) , 1 2 ( - n - ν + 2 ) , 1 - ν ; 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["1", "-", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Nu]", "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["1", "-", "\[Nu]"]], 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 n + 2 ν - 2 3 π ν 4 π 3 / 2 csc ( π ν ) Γ ( 1 - ν ) z - n - ν 2 F ~ 3 ( 1 - ν 2 , 1 - ν 2 ; 1 2 ( - n - ν + 1 ) , 1 2 ( - n - ν + 2 ) , 1 - ν ; - 1 4 ( z 2 ) ) 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["1", "-", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Nu]", "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["1", "-", "\[Nu]"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + 2 n - 2 ν - 2 3 π ν 4 π 3 / 2 ( - + cot ( π ν ) ) Γ ( ν + 1 ) z ν - n 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 n - 2 ν - 2 1 4 ( - 3 ) π ν π 3 / 2 ( + cot ( π ν ) ) Γ ( ν + 1 ) z ν - n 2 F ~ 3 ( ν + 1 2 , ν + 2 2 ; 1 2 ( - n + ν + 1 ) , 1 2 ( - n + ν + 2 ) , ν + 1 ; - 1 4 ( z 2 ) ) 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[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] /; ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition z n KelvinKei ν z -1 2 n 2 ν -2 1 4 -3 ν 3 2 ν Gamma 1 -1 ν z -1 n -1 ν HypergeometricPFQRegularized 1 -1 ν 2 -1 1 -1 ν 2 -1 1 2 -1 n -1 ν 1 1 2 -1 n -1 ν 2 1 -1 ν z 2 4 -1 2 n 2 ν -2 3 ν 4 -1 3 2 ν Gamma 1 -1 ν z -1 n -1 ν HypergeometricPFQRegularized 1 -1 ν 2 -1 1 -1 ν 2 -1 1 2 -1 n -1 ν 1 1 2 -1 n -1 ν 2 1 -1 ν -1 1 4 z 2 2 n -1 2 ν -2 3 ν 4 -1 3 2 -1 ν Gamma ν 1 z ν -1 n 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 -1 2 n -1 2 ν -2 1 4 -3 ν 3 2 ν Gamma ν 1 z ν -1 n HypergeometricPFQRegularized ν 1 2 -1 ν 2 2 -1 1 2 -1 n ν 1 1 2 -1 n ν 2 ν 1 -1 1 4 z 2 ν n [/itex]

 Rule Form

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

 2007-05-02

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