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 KelvinKei

 http://functions.wolfram.com/03.19.06.0016.01

 Input Form

 KelvinKei[\[Nu], z] == ((-2^(-1 + \[Nu])) Sin[(3 Pi \[Nu])/4] Gamma[\[Nu]] Sum[((-1)^k (z/4)^(4 k))/(Pochhammer[(1 - \[Nu])/2, k] Pochhammer[1 - \[Nu]/2, k] Pochhammer[1/2, k] k!), {k, 0, Infinity}])/ z^\[Nu] - 2^(-1 - \[Nu]) z^\[Nu] Sin[(Pi \[Nu])/4] Gamma[-\[Nu]] Sum[((-1)^k (z/4)^(4 k))/(Pochhammer[(\[Nu] + 1)/2, k] Pochhammer[1 + \[Nu]/2, k] Pochhammer[1/2, k] k!), {k, 0, Infinity}] - 2^(-3 + \[Nu]) z^(2 - \[Nu]) Cos[(3 Pi \[Nu])/4] Gamma[\[Nu] - 1] Sum[((-1)^k (z/4)^(4 k))/(Pochhammer[1 - \[Nu]/2, k] Pochhammer[(3 - \[Nu])/2, k] Pochhammer[3/2, k] k!), {k, 0, Infinity}] - 2^(-3 - \[Nu]) z^(2 + \[Nu]) Cos[(Pi \[Nu])/4] Gamma[-1 - \[Nu]] Sum[((-1)^k (z/4)^(4 k))/(Pochhammer[1 + \[Nu]/2, k] Pochhammer[(3 + \[Nu])/2, k] Pochhammer[3/2, k] k!), {k, 0, Infinity}] /; !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], RowBox[List["Gamma", "[", "\[Nu]", "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "4"], "]"]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Nu]"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "-", "1"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Nu]"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "4"], "]"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 kei ν ( z ) - 2 ν - 1 Γ ( ν ) sin ( 3 π ν 4 ) z - ν k = 0 ( - 1 ) k ( z 4 ) 4 k ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! - 2 - ν - 1 sin ( π ν 4 ) Γ ( - ν ) z ν k = 0 ( - 1 ) k ( z 4 ) 4 k ( ν + 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ")"]], "k"], Pochhammer] ( ν 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! - 2 ν - 3 cos ( 3 π ν 4 ) Γ ( ν - 1 ) z 2 - ν k = 0 ( - 1 ) k ( z 4 ) 4 k ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ")"]], "k"], Pochhammer] ( 3 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["3", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! - 2 - ν - 3 cos ( π ν 4 ) Γ ( - ν - 1 ) z ν + 2 k = 0 ( - 1 ) k ( z 4 ) 4 k ( ν 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( ν + 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[Nu]", "+", "3"]], "2"], ")"]], "k"], Pochhammer] ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] k ! /; ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition KelvinKei ν z -1 2 ν -1 Gamma ν 3 ν 4 -1 z -1 ν k 0 -1 k z 4 -1 4 k Pochhammer 1 -1 ν 2 -1 k Pochhammer 1 -1 ν 2 -1 k Pochhammer 1 2 k k -1 -1 2 -1 ν -1 ν 4 -1 Gamma -1 ν z ν k 0 -1 k z 4 -1 4 k Pochhammer ν 1 2 -1 k Pochhammer ν 2 -1 1 k Pochhammer 1 2 k k -1 -1 2 ν -3 3 ν 4 -1 Gamma ν -1 z 2 -1 ν k 0 -1 k z 4 -1 4 k Pochhammer 1 -1 ν 2 -1 k Pochhammer 3 -1 ν 2 -1 k Pochhammer 3 2 k k -1 -1 2 -1 ν -3 ν 4 -1 Gamma -1 ν -1 z ν 2 k 0 -1 k z 4 -1 4 k Pochhammer ν 2 -1 1 k Pochhammer ν 3 2 -1 k Pochhammer 3 2 k k -1 ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Nu]"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "-", "1"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Nu]"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "4"], ")"]], RowBox[List["4", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02