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; }

 Erfc

 http://functions.wolfram.com/06.27.21.0031.01

 Input Form

 Integrate[z^n Sin[b z] Erfc[a z], z] == (1/2) b^(-1 - n) ((-I^n) Erfc[a z] (Gamma[1 + n, (-I) b z] + (-1)^n Gamma[1 + n, I b z]) + ((a n!)/Sqrt[Pi]) Exp[-(b^2/(4 a^2))] ((-I)^n Sum[((I b)^m/m!) (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-(((-I) b)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z - (I b)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z - (I b)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z - (I b)/(2 Sqrt[-a^2]))^2], {k, 0, m}], {m, 0, n}] + I^n Sum[(((-I) b)^m/m!) (-a^2)^((1/2) (-1 - m)) Sum[Binomial[m, k] (-((I b)/(2 Sqrt[-a^2])))^(m - k) (Sqrt[-a^2] z + (I b)/(2 Sqrt[-a^2]))^(1 + k) (-(Sqrt[-a^2] z + (I b)/(2 Sqrt[-a^2]))^2)^((1/2) (-1 - k)) Gamma[(1 + k)/2, -(Sqrt[-a^2] z + (I b)/(2 Sqrt[-a^2]))^2], {k, 0, m}], {m, 0, n}])) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ImaginaryI]", "n"]]], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]]]], "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["a", " ", RowBox[List["n", "!"]]]], SqrtBox["\[Pi]"]], RowBox[List["Exp", "[", RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b"]], ")"]], "m"], RowBox[List["m", "!"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["a", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "m"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], RowBox[List["m", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], RowBox[List["1", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]]]], "]"]]]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], ")"]], "m"], RowBox[List["m", "!"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["a", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "m"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], RowBox[List["m", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], RowBox[List["1", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]]]], "]"]]]]]]]]]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n sin ( b z ) erfc ( a z ) z 1 2 b - n - 1 ( - n erfc ( a z ) ( Γ ( n + 1 , - b z ) + ( - 1 ) n Γ ( n + 1 , b z ) ) + a n ! π exp ( - b 2 4 a 2 ) ( ( - ) n m = 0 n 1 m ! ( b ) m ( - a 2 ) 1 2 ( - m - 1 ) k = 0 m ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - - b 2 - a 2 ) m - k ( - a 2 z - b 2 - a 2 ) k + 1 ( - ( - a 2 z - b 2 - a 2 ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( - a 2 z - b 2 - a 2 ) 2 ) + n m = 0 n 1 m ! ( - b ) m ( - a 2 ) 1 2 ( - m - 1 ) k = 0 m ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - b 2 - a 2 ) m - k ( - a 2 z + b 2 - a 2 ) k + 1 ( - ( - a 2 z + b 2 - a 2 ) 2 ) 1 2 ( - k - 1 ) Γ ( k + 1 2 , - ( - a 2 z + b 2 - a 2 ) 2 ) ) ) /; n Condition z z n b z Erfc a z 1 2 b -1 n -1 -1 n Erfc a z Gamma n 1 -1 b z -1 n Gamma n 1 b z a n 1 2 -1 -1 b 2 4 a 2 -1 -1 n m 0 n 1 m -1 b m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k -1 -1 b 2 -1 a 2 1 2 -1 m -1 k -1 a 2 1 2 z -1 b 2 -1 a 2 1 2 -1 k 1 -1 -1 a 2 1 2 z -1 b 2 -1 a 2 1 2 -1 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 -1 a 2 1 2 z -1 b 2 -1 a 2 1 2 -1 2 n m 0 n 1 m -1 -1 b m -1 a 2 1 2 -1 m -1 k 0 m Binomial m k -1 b 2 -1 a 2 1 2 -1 m -1 k -1 a 2 1 2 z b 2 -1 a 2 1 2 -1 k 1 -1 -1 a 2 1 2 z b 2 -1 a 2 1 2 -1 2 1 2 -1 k -1 Gamma k 1 2 -1 -1 -1 a 2 1 2 z b 2 -1 a 2 1 2 -1 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", RowBox[List["Sin", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ImaginaryI]", "n"]]], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b", " ", "z"]]]], "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", " ", RowBox[List["n", "!"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "b"]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["a", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "m"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], RowBox[List["m", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], RowBox[List["1", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]]]], "]"]]]]]]]], RowBox[List["m", "!"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["a", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "m"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], RowBox[List["m", "-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], RowBox[List["1", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "k"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "k"]], "2"], ",", RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]], " ", "z"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["a", "2"]]]]]]]]], ")"]], "2"]]]]], "]"]]]]]]]], RowBox[List["m", "!"]]]]]]]]], ")"]]]], SqrtBox["\[Pi]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29