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 KelvinKei

 http://functions.wolfram.com/03.19.06.0021.01

 Input Form

 KelvinKei[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /; Subscript[F, n][z, \[Nu]] == (-((z^\[Nu] Gamma[-\[Nu]])/2^(\[Nu] + 1))) Sum[(Sin[(1/4) Pi (\[Nu] - 2 k)]/(Pochhammer[\[Nu] + 1, k] k!)) (z/2)^(2 k), {k, 0, n}] - (Gamma[\[Nu]]/(z^\[Nu] 2^(-\[Nu] + 1))) Sum[(Sin[(1/4) Pi (3 \[Nu] - 2 k)]/(Pochhammer[1 - \[Nu], k] k!)) (z/2)^(2 k), {k, 0, n}] == KelvinKei[\[Nu], z] - ((-I)^n 2^(-4 - 2 n + \[Nu]) E^((3 I Pi \[Nu])/4) Pi z^(2 + 2 n - \[Nu]) Csc[Pi \[Nu]] HypergeometricPFQRegularized[{1}, {2 + n, 2 + n - \[Nu]}, -((I z^2)/4)] + (I^n 2^(-4 - 2 n + \[Nu]) Pi z^(2 + 2 n - \[Nu]) Csc[Pi \[Nu]] HypergeometricPFQRegularized[{1}, {2 + n, 2 + n - \[Nu]}, (I z^2)/4])/E^((3/4) I Pi \[Nu]) - (-I)^n 2^(-4 - 2 n - \[Nu]) E^((I Pi \[Nu])/4) Pi z^(2 + 2 n + \[Nu]) Csc[Pi \[Nu]] HypergeometricPFQRegularized[{1}, {2 + n, 2 + n + \[Nu]}, -((I z^2)/4)] - (I^n 2^(-4 - 2 n - \[Nu]) Pi z^(2 + 2 n + \[Nu]) Csc[Pi \[Nu]] HypergeometricPFQRegularized[{1}, {2 + n, 2 + n + \[Nu]}, (I z^2)/4])/E^((1/4) I Pi \[Nu])) && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " "]], SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "+", "1"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "]"]], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "-", RowBox[List["(", 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 MathML Form

 kei ν ( z ) F ( z , ν ) /; ( ( F n ( z , ν ) - z ν Γ ( - ν ) 2 ν + 1 k = 0 n sin ( 1 4 π ( ν - 2 k ) ) ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( z 2 ) 2 k - z - ν Γ ( ν ) 2 1 - ν k = 0 n sin ( 1 4 π ( 3 ν - 2 k ) ) ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] k ! ( z 2 ) 2 k kei ν ( z ) - ( ( - ) n 2 - 2 n + ν - 4 3 π ν 4 π z 2 n - ν + 2 csc ( π ν ) 1 F ~ 2 ( 1 ; n + 2 , n - ν + 2 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", 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["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "-", "\[Nu]", "+", "2"]], 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 2 - 2 n - ν - 4 π ν 4 π z 2 n + ν + 2 csc ( π ν ) 1 F ~ 2 ( 1 ; n + 2 , n + ν + 2 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", 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["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "\[Nu]", "+", "2"]], 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] + 2 - 2 n + ν - 4 - 3 π ν 4 n π z 2 n - ν + 2 csc ( π ν ) 1 F ~ 2 ( 1 ; n + 2 , n - ν + 2 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", 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["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "-", "\[Nu]", "+", "2"]], 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] - n 2 - 2 n - ν - 4 - π ν 4 π z 2 n + ν + 2 csc ( π ν ) 1 F ~ 2 ( 1 ; n + 2 , n + ν + 2 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", 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["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "\[Nu]", "+", "2"]], 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] ) ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) Condition KelvinKei ν z Subscript F z ν Subscript F n z ν -1 z ν Gamma -1 ν 2 ν 1 -1 k 0 n 1 4 ν -1 2 k Pochhammer ν 1 k k -1 z 2 -1 2 k -1 z -1 ν Gamma ν 2 1 -1 ν -1 k 0 n 1 4 3 ν -1 2 k Pochhammer 1 -1 ν k k -1 z 2 -1 2 k KelvinKei ν z -1 -1 n 2 -2 n ν -4 3 ν 4 -1 z 2 n -1 ν 2 ν HypergeometricPFQRegularized 1 n 2 n -1 ν 2 -1 z 2 4 -1 -1 -1 n 2 -2 n -1 ν -4 ν 4 -1 z 2 n ν 2 ν HypergeometricPFQRegularized 1 n 2 n ν 2 -1 z 2 4 -1 2 -2 n ν -4 -1 3 ν 4 -1 n z 2 n -1 ν 2 ν HypergeometricPFQRegularized 1 n 2 n -1 ν 2 z 2 4 -1 -1 n 2 -2 n -1 ν -4 -1 ν 4 -1 z 2 n ν 2 ν HypergeometricPFQRegularized 1 n 2 n ν 2 z 2 4 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], SuperscriptBox["2", RowBox[List["\[Nu]", "+", "1"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Nu]"]], "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "+", "1"]]]]]], "\[Equal]", RowBox[List[RowBox[List["KelvinKei", "[", 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RowBox[List["2", " ", "n"]], "-", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "n"]], "+", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], ")"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], "-", RowBox[List["2", " ", "n"]], "-", "\[Nu]"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "n"]], "+", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n", "+", "\[Nu]"]]]], "}"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]], "]"]]]]]], ")"]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02