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 WhittakerW

 http://functions.wolfram.com/07.45.06.0019.01

 Input Form

 WhittakerW[\[Nu], \[Mu], z] == (-(z^(1/2 + \[Mu])/Gamma[1/2 + \[Mu] - \[Nu]])) ((Log[z]/((-z)^(2 \[Mu]) (-2 \[Mu])!)) Sum[(Pochhammer[1/2 - \[Mu] - \[Nu], j]/(j! (k - j)! Pochhammer[1 - 2 \[Mu], j])) (-(1/2))^(k - j) z^k, {k, 0, Infinity}, {j, 0, k}] - (1/Pochhammer[1/2 + \[Mu] - \[Nu], -2 \[Mu]]) Sum[(((-1)^(k - j) (j - k - 2 \[Mu] - 1)! Pochhammer[ 1/2 + \[Mu] - \[Nu], k - j])/((k - j)! j!)) (-(1/2))^j z^k, {k, 0, -1 - 2 \[Mu]}, {j, 0, k}] + (1/(z^(2 \[Mu]) Pochhammer[1/2 + \[Mu] - \[Nu], -2 \[Mu]])) Sum[(((-1)^(2 \[Mu] + j) (-(1/2))^(j + k + 1) j! Pochhammer[1/2 + \[Mu] - \[Nu], -1 - 2 \[Mu] - j])/ ((-1 - 2 \[Mu] - j)! (j + k + 1)!)) z^k, {k, 0, Infinity}, {j, 0, -1 - 2 \[Mu]}] - Sum[(((-(1/2))^(k - j) Pochhammer[1/2 - \[Mu] - \[Nu], j])/ (j! (j - 2 \[Mu])! (k - j)!)) (PolyGamma[1 + j] + PolyGamma[1 + j - 2 \[Mu]] - PolyGamma[1/2 + j - \[Mu] - \[Nu]]) z^k, {k, 0, Infinity}, {j, 0, k}]/(-z)^(2 \[Mu])) /; Element[-2 \[Mu], Integers] && -2 \[Mu] > 0

 Standard Form

 Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]], RowBox[List["Log", "[", "z", "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]], RowBox[List[RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ",", "j"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], SuperscriptBox["z", "k"]]]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Mu]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "j"]]], RowBox[List[RowBox[List["(", RowBox[List["j", "-", "k", "-", RowBox[List["2", " ", "\[Mu]"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List["k", "-", "j"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], RowBox[List["j", "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "j"], SuperscriptBox["z", "k"]]]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "\[Mu]"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Mu]"]]]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "j"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "+", "k", "+", "1"]]], RowBox[List["j", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Mu]"]], "-", "j"]]]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Mu]"]], "-", "j"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k", "+", "1"]], ")"]], "!"]]]]], SuperscriptBox["z", "k"]]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], 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RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], "\[Mu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "\[Mu]"]], ">", "0"]]]]]]]]]]

 MathML Form

 W TagBox["W", WhittakerW] ν , μ ( z ) - z μ + 1 2 Γ ( μ - ν + 1 2 ) ( - ( - z ) - 2 μ k = 0 j = 0 k ( - 1 2 ) k - j ( 1 2 - μ - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] j ! ( j - 2 μ ) ! ( k - j ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( j + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( j - 2 μ + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j - μ - ν + 1 2 ) ) z k + ( - z ) - 2 μ log ( z ) ( - 2 μ ) ! k = 0 j = 0 k ( - μ - ν + 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] j ! ( 1 - 2 μ ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Mu]"]]]], ")"]], "j"], Pochhammer] ( k - j ) ! ( - 1 2 ) k - j z k - 1 ( μ - ν + 1 2 ) - 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]], Pochhammer] k = 0 - 2 μ - 1 j = 0 k ( - 1 ) k - j ( j - k - 2 μ - 1 ) ! ( μ - ν + 1 2 ) k - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], Pochhammer] ( k - j ) ! j ! ( - 1 2 ) j z k + z - 2 μ ( μ - ν + 1 2 ) - 2 μ TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]], Pochhammer] k = 0 j = 0 - 2 μ - 1 ( - 1 ) j + 2 μ j ! ( μ - ν + 1 2 ) - j - 2 μ - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "-", RowBox[List["2", " ", "\[Mu]"]], "-", "1"]]], Pochhammer] ( - j - 2 μ - 1 ) ! ( j + k + 1 ) ! ( - 1 2 ) j + k + 1 z k ) /; - 2 μ + Condition WhittakerW ν μ z -1 z μ 1 2 Gamma μ -1 ν 1 2 -1 -1 -1 z -2 μ j 0 k k 0 -1 1 2 k -1 j Pochhammer 1 2 -1 μ -1 ν j j j -1 2 μ k -1 j -1 PolyGamma j 1 PolyGamma j -1 2 μ 1 -1 PolyGamma j -1 μ -1 ν 1 2 z k -1 z -2 μ z -2 μ -1 j 0 k k 0 Pochhammer -1 μ -1 ν 1 2 j j Pochhammer 1 -1 2 μ j k -1 j -1 -1 1 2 k -1 j z k -1 1 Pochhammer μ -1 ν 1 2 -2 μ -1 j 0 k k 0 -2 μ -1 -1 k -1 j j -1 k -1 2 μ -1 Pochhammer μ -1 ν 1 2 k -1 j k -1 j j -1 -1 1 2 j z k z -2 μ Pochhammer μ -1 ν 1 2 -2 μ -1 j 0 -2 μ -1 k 0 -1 j 2 μ j Pochhammer μ -1 ν 1 2 -1 j -1 2 μ -1 -1 j -1 2 μ -1 j k 1 -1 -1 1 2 j k 1 z k -2 μ SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "\[Mu]"]]], " ", RowBox[List["Log", "[", "z", "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Mu]", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], RowBox[List["k", "-", "j"]]], " ", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02