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 WhittakerM

 http://functions.wolfram.com/07.44.03.0063.01

 Input Form

 WhittakerM[-((2 m + 1)/4) + n, -((2 m + 1)/4), z] == ((-1)^m ((2 m + 1)/2) z^((5 + 2 m)/4) Sum[((-1)^k/k!) LaguerreL[-1 - k - m + n, k, z] ((Erfi[Sqrt[z]] Gamma[-(1/2) + k - m])/Sqrt[z] - E^z Sum[(-z)^j/Pochhammer[-(1/2) + k - m, 2 + j - k + m], {j, 0, -2 + k - m}] + E^z Sum[(-z)^(-j - 1)/Pochhammer[ -(1/2) + k - m, 1 - j - k + m], {j, 0, -k + m}]), {k, 0, -1 - m + n}])/E^(z/2) /; Element[n, Integers] && Element[m, Integers] && n > m

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", "m"]], "+", "1"]], "4"]]], "+", "n"]], ",", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", "m"]], "+", "1"]], "4"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], FractionBox[RowBox[List[RowBox[List["2", "m"]], "+", "1"]], "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "/", "2"]]], " ", SuperscriptBox["z", FractionBox[RowBox[List["5", "+", RowBox[List["2", "m"]]]], "4"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "-", "m", "+", "n"]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["k", "!"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "k", "-", "m", "+", "n"]], ",", "k", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Erfi", "[", SqrtBox["z"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "-", "m"]], "]"]]]], SqrtBox["z"]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "2"]], "+", "k", "-", "m"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "-", "m"]], ",", RowBox[List["2", "+", "j", "-", "k", "+", "m"]]]], "]"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "k"]], "+", "m"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "-", "1"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "-", "m"]], ",", RowBox[List["1", "-", "j", "-", "k", "+", "m"]]]], "]"]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "m"]]]]]]]]

 MathML Form

 M TagBox["M", WhittakerM] n - 2 m + 1 4 , - 2 m + 1 4 ( z ) ( - 1 ) m 2 m + 1 2 - z 2 z 2 m + 5 4 k = 0 n - m - 1 ( - 1 ) k k ! L - k - m + n - 1 k ( z ) ( erfi ( z ) Γ ( k - m - 1 2 ) z - z j = 0 k - m - 2 ( - z ) j ( k - m - 1 2 ) j - k + m + 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "m", "-", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "k", "+", "m", "+", "2"]]], Pochhammer] + z j = 0 m - k ( - z ) - j - 1 ( k - m - 1 2 ) - j - k + m + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "m", "-", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["-", "j"]], "-", "k", "+", "m", "+", "1"]]], Pochhammer] ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] m TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n > m Condition WhittakerM n -1 2 m 1 4 -1 -1 2 m 1 4 -1 z -1 m 2 m 1 2 -1 -1 z 2 -1 z 2 m 5 4 -1 k 0 n -1 m -1 -1 k k -1 LaguerreL -1 k -1 m n -1 k z Erfi z 1 2 Gamma k -1 m -1 1 2 z 1 2 -1 -1 z j 0 k -1 m -2 -1 z j Pochhammer k -1 m -1 1 2 j -1 k m 2 -1 z j 0 m -1 k -1 z -1 j -1 Pochhammer k -1 m -1 1 2 -1 j -1 k m 1 -1 n m n m [/itex]

 Rule Form

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

 2007-05-02