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; }

 Hypergeometric1F1

 http://functions.wolfram.com/07.20.03.0131.01

 Input Form

 Hypergeometric1F1[n, 1/2 + m, z] == (Sqrt[Pi]/2) Pochhammer[3/2, -1 + m] z^(1/2 - m) E^z Erf[Sqrt[z]] Sum[Binomial[-m + n, k] LaguerreL[-1 + k + m, 1/2 - k - m, -z], {k, 0, -m + n}] + (1/2) z^(1 - m) Pochhammer[3/2, -1 + m] Sum[Binomial[-m + n, k] Sum[(1/p) LaguerreL[-1 + k + m - p, 1/2 - k - m + p, -z] LaguerreL[-1 + p, 1/2 - p, z], {p, 1, -1 + k + m}], {k, 0, -m + n}] /; Element[n, Integers] && n > 0 && Element[m, Integers] && m > 0 && m <= n

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric1F1", "[", RowBox[List["n", ",", RowBox[List[FractionBox["1", "2"], "+", "m"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SqrtBox["\[Pi]"], "2"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]], "]"]], SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", "z"], RowBox[List["Erf", "[", SqrtBox["z"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "m"]], "+", "n"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", "m"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", RowBox[List["1", "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "m"]], "+", "n"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", "m"]]], RowBox[List[FractionBox["1", "p"], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", "m", "-", "p"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "-", "m", "+", "p"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "p"]], ",", RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", "z"]], "]"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]

 MathML Form

 1 F 1 ( n ; m + 1 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["n", Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["m", "+", FractionBox["1", "2"]]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric1F1] 1 2 ( 3 2 ) m - 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["m", "-", "1"]]], Pochhammer] z 1 - m k = 0 n - m ( n - m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] p = 1 k + m - 1 1 p L k + m - p - 1 - k - m + p + 1 2 ( - z ) L p - 1 1 2 - p ( z ) + π 2 ( 3 2 ) m - 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["m", "-", "1"]]], Pochhammer] z 1 2 - m z erf ( z ) k = 0 n - m ( n - m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] L k + m - 1 - k - m + 1 2 ( - z ) /; n + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] m + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[List[], Integers]] m n Condition Hypergeometric1F1 n m 1 2 z 1 2 Pochhammer 3 2 m -1 z 1 -1 m k 0 n -1 m Binomial n -1 m k p 1 k m -1 1 p -1 LaguerreL k m -1 p -1 -1 k -1 m p 1 2 -1 z LaguerreL p -1 1 2 -1 p z 1 2 2 -1 Pochhammer 3 2 m -1 z 1 2 -1 m z Erf z 1 2 k 0 n -1 m Binomial n -1 m k LaguerreL k m -1 -1 k -1 m 1 2 -1 z n m m n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric1F1", "[", RowBox[List["n_", ",", RowBox[List[FractionBox["1", "2"], "+", "m_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]], "]"]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "m"]]], " ", SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["Erf", "[", SqrtBox["z"], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "m"]], "+", "n"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", "m"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", RowBox[List["1", "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "m"]], "+", "n"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", "m"]]], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "k", "+", "m", "-", "p"]], ",", RowBox[List[FractionBox["1", "2"], "-", "k", "-", "m", "+", "p"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "p"]], ",", RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", "z"]], "]"]]]], "p"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["m", "\[LessEqual]", "n"]]]]]]]]]]

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

 2007-05-02