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/06.27.21.0046.01

 Input Form

 Integrate[z Cos[b z^2] Erfc[c + a z], z] == (1/(b (a^4 + b^2))) (((1/8 - I/8) (Sqrt[2] a ((-I) a^2 + b) Sqrt[I a^2 + b] E^((2 I a^2 b c^2)/(a^4 + b^2)) Erf[((1 + I) (a c + a^2 z - I b z))/ (Sqrt[2] Sqrt[I a^2 + b])] + (a^2 - I b) (I Sqrt[2] a Sqrt[(-I) a^2 + b] Erfi[((1 + I) (a c + a^2 z + I b z))/ (Sqrt[2] Sqrt[(-I) a^2 + b])] + (2 + 2 I) (a^2 + I b) E^((I b c^2)/(a^2 + I b)) Erfc[c + a z] Sin[b z^2])))/ E^((I b c^2)/(a^2 + I b)))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List["z", " ", RowBox[List["Cos", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["Erfc", "[", RowBox[List["c", "+", RowBox[List["a", " ", "z"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "2"]]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "8"], "-", FractionBox["\[ImaginaryI]", "8"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["c", "2"]]], RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"]]], "+", "b"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "b", " ", SuperscriptBox["c", "2"]]], RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "2"]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], ")"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"]]], "+", "b"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "a", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "+", "b"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], ")"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "+", "b"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["c", "2"]]], RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", RowBox[List["Erfc", "[", RowBox[List["c", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z cos ( b z 2 ) erfc ( c + a z ) z 1 b ( a 4 + b 2 ) ( ( 1 - 8 ) exp ( - b c 2 a 2 + b ) ( 2 a ( b - a 2 ) a 2 + b exp ( 2 a 2 b c 2 a 4 + b 2 ) erf ( ( 1 + ) ( z a 2 + c a - b z ) 2 a 2 + b ) + ( a 2 - b ) ( 2 a b - a 2 erfi ( ( 1 + ) ( z a 2 + c a + b z ) 2 b - a 2 ) + ( a 2 + b ) exp ( b c 2 a 2 + b ) ( 2 + 2 ) erfc ( c + a z ) sin ( b z 2 ) ) ) ) z z b z 2 Erfc c a z 1 b a 4 b 2 -1 1 -1 8 -1 -1 b c 2 a 2 b -1 2 1 2 a b -1 a 2 a 2 b 1 2 2 a 2 b c 2 a 4 b 2 -1 Erf 1 z a 2 c a -1 b z 2 1 2 a 2 b 1 2 -1 a 2 -1 b 2 1 2 a b -1 a 2 1 2 Erfi 1 z a 2 c a b z 2 1 2 b -1 a 2 1 2 -1 a 2 b b c 2 a 2 b -1 2 2 Erfc c a z b z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["z_", " ", RowBox[List["Cos", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["c_", "+", RowBox[List["a_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "8"], "-", FractionBox["\[ImaginaryI]", "8"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["c", "2"]]], RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"]]], "+", "b"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "b", " ", SuperscriptBox["c", "2"]]], RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "2"]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], ")"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"]]], "+", "b"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", "a", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "+", "b"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]]], ")"]]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "+", "b"]]]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", RowBox[List["2", " ", "\[ImaginaryI]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "b", " ", SuperscriptBox["c", "2"]]], RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", RowBox[List["Erfc", "[", RowBox[List["c", "+", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "2"]]], ")"]]]]]]]]]

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

 2001-10-29