html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 WhittakerM

 http://functions.wolfram.com/07.44.03.0061.01

 Input Form

 WhittakerM[-((2 m - 1)/4) + n, -((2 m - 1)/4), z] == (((-1)^m z^((3 - 2 m)/4) m!)/(E^(z/2) (2 n! Pochhammer[-(1/2), m]))) (((Sqrt[Pi] Erfi[Sqrt[z]])/Sqrt[z]) Sum[((k + n)!/k!) LaguerreL[-k + m, 1/2 + k - m, -z] LaguerreL[k + n, -(1/2) - k, z], {k, 0, m}] + E^z Sum[((k + n)!/k!) LaguerreL[-k + m, 1/2 + k - m, -z] Sum[(1/p) LaguerreL[k + n - p, -(1/2) - k + p, z] LaguerreL[-1 + p, 1/2 - p, -z], {p, 1, k + n}], {k, 0, m}]) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0

 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[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "/", "2"]]], " ", SuperscriptBox["z", FractionBox[RowBox[List["3", "-", RowBox[List["2", "m"]]]], "4"]], " ", RowBox[List["m", "!"]]]], RowBox[List["2", " ", RowBox[List["n", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "m"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", SqrtBox["z"], "]"]]]], SqrtBox["z"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]], " "]], RowBox[List["k", "!"]]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "m"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "n"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], ",", "z"]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]], " "]], RowBox[List["k", "!"]]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "m"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["k", "+", "n"]]], RowBox[List[FractionBox["1", "p"], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "n", "-", "p"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k", "+", "p"]], ",", "z"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "p"]], ",", RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 M TagBox["M", WhittakerM] n - 2 m - 1 4 , - 2 m - 1 4 ( z ) ( - 1 ) m - z 2 z 3 - 2 m 4 m ! 2 n ! ( - 1 2 ) m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "m"], Pochhammer] ( π erfi ( z ) z k = 0 m ( k + n ) ! k ! L m - k k - m + 1 2 ( - z ) L k + n - k - 1 2 ( z ) + z k = 0 m ( k + n ) ! k ! L m - k k - m + 1 2 ( - z ) p = 1 k + n 1 p L k + n - p p - k - 1 2 ( z ) L p - 1 1 2 - p ( - z ) ) /; n m Condition WhittakerM n -1 2 m -1 4 -1 -1 2 m -1 4 -1 z -1 m -1 z 2 -1 z 3 -1 2 m 4 -1 m 2 n Pochhammer -1 1 2 m -1 1 2 Erfi z 1 2 z 1 2 -1 k 0 m k n k -1 LaguerreL m -1 k k -1 m 1 2 -1 z LaguerreL k n -1 k -1 1 2 z z k 0 m k n k -1 LaguerreL m -1 k k -1 m 1 2 -1 z p 1 k n 1 p -1 LaguerreL k n -1 p p -1 k -1 1 2 z LaguerreL p -1 1 2 -1 p -1 z n m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerM", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m_"]], "-", "1"]], ")"]]]], "+", "n_"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m_"]], "-", "1"]], ")"]]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", "2"]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "m"]]]], ")"]]]]], " ", RowBox[List["m", "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", SqrtBox["z"], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "m"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "n"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], ",", "z"]], "]"]]]], RowBox[List["k", "!"]]]]]]], SqrtBox["z"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "m"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["k", "+", "n"]]], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "n", "-", "p"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k", "+", "p"]], ",", "z"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "p"]], ",", RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", RowBox[List["-", "z"]]]], "]"]]]], "p"]]]]], RowBox[List["k", "!"]]]]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["n", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "m"]], "]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2006)

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

 2007-05-02