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; }

 HypergeometricU

 http://functions.wolfram.com/07.33.03.0066.01

 Input Form

 HypergeometricU[n + 1/2, m + 1/2, z] == (-1)^(m + n) Gamma[1/2 - n] E^z LaguerreL[-m + n, -(1/2) + m, -z] + (((-1)^(-1 + m) 2^(2 n) m! z^(1/2 - m))/(2 n)!) ((Sqrt[Pi]/Sqrt[z]) E^z Erf[Sqrt[z]] Sum[((k + n)!/k!) LaguerreL[-k + m, 1/2 + k - m, z] LaguerreL[k + n, -(1/2) - k, -z], {k, 0, m}] + 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["HypergeometricU", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", RowBox[List["m", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", "z"], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "+", "n"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "m"]], ",", RowBox[List["-", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], " ", SuperscriptBox["2", RowBox[List["2", " ", "n"]]], " ", RowBox[List["m", "!"]], SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "m"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List["2", "n"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SqrtBox["\[Pi]"], SqrtBox["z"]], " ", SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["Erf", "[", SqrtBox["z"], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[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"]], ",", "z"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "n"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], ",", RowBox[List["-", "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"]], ",", "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"]], ",", 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", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 U TagBox["U", HypergeometricU] ( n + 1 2 , m + 1 2 , z ) ( - 1 ) m + n Γ ( 1 2 - n ) z L n - m m - 1 2 ( - z ) + ( - 1 ) m - 1 2 2 n m ! z 1 2 - m ( 2 n ) ! ( π z z erf ( z ) k = 0 m ( k + n ) ! k ! L m - k k - m + 1 2 ( z ) L k + n - k - 1 2 ( - 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 - k + p - 1 2 ( - z ) L p - 1 1 2 - p ( z ) ) /; n m Condition HypergeometricU n 1 2 m 1 2 z -1 m n Gamma 1 2 -1 n z LaguerreL n -1 m m -1 1 2 -1 z -1 m -1 2 2 n m z 1 2 -1 m 2 n -1 1 2 z 1 2 -1 z Erf z 1 2 k 0 m k n k -1 LaguerreL m -1 k k -1 m 1 2 z LaguerreL k n -1 k -1 1 2 -1 z k 0 m k n k -1 LaguerreL m -1 k k -1 m 1 2 z p 1 k n 1 p -1 LaguerreL k n -1 p -1 k p -1 1 2 -1 z LaguerreL p -1 1 2 -1 p z n m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["n_", "+", FractionBox["1", "2"]]], ",", RowBox[List["m_", "+", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "+", "n"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "m"]], ",", RowBox[List["-", "z"]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "m"]]], " ", SuperscriptBox["2", RowBox[List["2", " ", "n"]]], " ", RowBox[List["m", "!"]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "m"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["Erf", "[", 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"]], ",", "z"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "n"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], ",", RowBox[List["-", "z"]]]], "]"]]]], RowBox[List["k", "!"]]]]]]], 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"]], ",", "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"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "p"]], ",", RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", "z"]], "]"]]]], "p"]]]]], RowBox[List["k", "!"]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02