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/07.44.03.0055.01

 Input Form

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

 Standard Form

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

 MathML Form

 M TagBox["M", WhittakerM] 2 m - 1 4 - n , 2 m - 1 4 ( z ) 2 n + 1 2 ( m - n - 1 ) ! ( n + 3 2 ) m - n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["m", "-", "n", "-", "1"]]], Pochhammer] - z 2 z 2 m + 1 4 ( ( - 1 ) n π z - n - 1 2 erfi ( z ) p = 0 m - n - 1 z - p ( m - n - 1 p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "-", "n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["p", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 1 2 ) n + p TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["n", "+", "p"]]], Pochhammer] - 2 z p = 0 m - n - 1 ( - 1 ) p ( m - n - 1 p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "-", "n", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["p", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] 2 n + 2 p + 1 k = 1 n + p z - k ( - n - p - 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "-", "p", "-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ) /; n m TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] m > n Condition WhittakerM 2 m -1 4 -1 -1 n 2 m -1 4 -1 z 2 n 1 2 m -1 n -1 -1 Pochhammer n 3 2 m -1 n -1 -1 z 2 -1 z 2 m 1 4 -1 -1 n 1 2 z -1 n -1 1 2 Erfi z 1 2 p 0 m -1 n -1 z -1 p Binomial m -1 n -1 p Pochhammer 1 2 n p -1 2 z p 0 m -1 n -1 -1 p Binomial m -1 n -1 p 2 n 2 p 1 -1 k 1 n p z -1 k Pochhammer -1 n -1 p -1 1 2 k n m m n [/itex]

 Rule Form

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

 Contributed by

 Brychkov Yu.A. (2006)

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

 2007-05-02