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/06.27.21.0043.01

 Input Form

 Integrate[z^2 Cos[b z] Erfc[a z], z] == (1/(8 a^4 b^3 Sqrt[Pi])) (E^(-(b^2/(4 a^2)) - I b z - a^2 z^2) (-8 a^3 b E^(b^2/(4 a^2)) - 2 a b^3 E^(b^2/(4 a^2)) - 8 a^3 b E^((1/4) b (b/a^2 + 8 I z)) - 2 a b^3 E^((1/4) b (b/a^2 + 8 I z)) - 4 I a^3 b^2 E^(b^2/(4 a^2)) z + 4 I a^3 b^2 E^((1/4) b (b/a^2 + 8 I z)) z - I (8 a^4 + 2 a^2 b^2 + b^4) E^(z (I b + a^2 z)) Sqrt[Pi] Erf[(I b + 2 a^2 z)/(2 a)] + 8 a^4 E^(z (I b + a^2 z)) Sqrt[Pi] Erfi[b/(2 a) + I a z] + 2 a^2 b^2 E^(z (I b + a^2 z)) Sqrt[Pi] Erfi[b/(2 a) + I a z] + b^4 E^(z (I b + a^2 z)) Sqrt[Pi] Erfi[b/(2 a) + I a z] + 8 a^4 E^(b^2/(4 a^2) + a^2 z^2) Sqrt[Pi] Erfc[a z] (-I + b z + E^(2 I b z) (I + b z) + b^2 E^(I b z) z^2 Sin[b z])))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["b", "3"], " ", SqrtBox["\[Pi]"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", SuperscriptBox["a", "3"], " ", "b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], "-", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "3"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], "-", RowBox[List["8", " ", SuperscriptBox["a", "3"], " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "b", " ", RowBox[List["(", RowBox[List[FractionBox["b", SuperscriptBox["a", "2"]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", "z"]]]], ")"]]]]]]], "-", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "b", " ", RowBox[List["(", RowBox[List[FractionBox["b", SuperscriptBox["a", "2"]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", "z"]]]], ")"]]]]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], " ", "z"]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "b", " ", RowBox[List["(", RowBox[List[FractionBox["b", SuperscriptBox["a", "2"]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", "z"]]]], ")"]]]]], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox["a", "4"]]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["b", "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "z"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "+", RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["b", " ", "z"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]], " ", SuperscriptBox["z", "2"], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 z 2 cos ( b z ) erfc ( a z ) z 1 8 a 4 b 3 π exp ( - b 2 4 a 2 - z b - a 2 z 2 ) ( 8 z ( z a 2 + b ) π erfi ( b 2 a + a z ) a 4 + 8 b 2 4 a 2 + a 2 z 2 π erfc ( a z ) ( b 2 b z sin ( b z ) z 2 + b z - + 2 b z ( + b z ) ) a 4 - 8 b b 2 4 a 2 a 3 - 8 b 1 4 b ( b a 2 + 8 z ) a 3 - 4 b 2 b 2 4 a 2 z a 3 + 4 b 2 exp ( 1 4 b ( b a 2 + 8 z ) ) z a 3 + 2 b 2 z ( z a 2 + b ) π erfi ( b 2 a + a z ) a 2 - 2 b 3 b 2 4 a 2 a - 2 b 3 exp ( 1 4 b ( b a 2 + 8 z ) ) a - ( 8 a 4 + 2 b 2 a 2 + b 4 ) z ( z a 2 + b ) π erf ( 2 z a 2 + b 2 a ) + b 4 z ( z a 2 + b ) π erfi ( b 2 a + a z ) ) z z 2 b z Erfc a z 1 8 a 4 b 3 1 2 -1 -1 b 2 4 a 2 -1 -1 z b -1 a 2 z 2 8 z z a 2 b 1 2 Erfi b 2 a -1 a z a 4 8 b 2 4 a 2 -1 a 2 z 2 1 2 Erfc a z b 2 b z b z z 2 b z -1 2 b z b z a 4 -1 8 b b 2 4 a 2 -1 a 3 -1 8 b 1 4 b b a 2 -1 8 z a 3 -1 4 b 2 b 2 4 a 2 -1 z a 3 4 b 2 1 4 b b a 2 -1 8 z z a 3 2 b 2 z z a 2 b 1 2 Erfi b 2 a -1 a z a 2 -1 2 b 3 b 2 4 a 2 -1 a -1 2 b 3 1 4 b b a 2 -1 8 z a -1 8 a 4 2 b 2 a 2 b 4 z z a 2 b 1 2 Erf 2 z a 2 b 2 a -1 b 4 z z a 2 b 1 2 Erfi b 2 a -1 a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", RowBox[List["Cos", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]], "-", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", SuperscriptBox["a", "3"], " ", "b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], "-", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "3"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]]], "-", RowBox[List["8", " ", SuperscriptBox["a", "3"], " ", "b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "b", " ", RowBox[List["(", RowBox[List[FractionBox["b", SuperscriptBox["a", "2"]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", "z"]]]], ")"]]]]]]], "-", RowBox[List["2", " ", "a", " ", SuperscriptBox["b", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "b", " ", RowBox[List["(", RowBox[List[FractionBox["b", SuperscriptBox["a", "2"]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", "z"]]]], ")"]]]]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]], " ", "z"]], "+", RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "b", " ", RowBox[List["(", RowBox[List[FractionBox["b", SuperscriptBox["a", "2"]], "+", RowBox[List["8", " ", "\[ImaginaryI]", " ", "z"]]]], ")"]]]]], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox["a", "4"]]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"]]], "+", SuperscriptBox["b", "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "z"]]]], RowBox[List["2", " ", "a"]]], "]"]]]], "+", RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox["b", RowBox[List["2", " ", "a"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]], "+", RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["b", " ", "z"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "b", " ", "z"]]], " ", SuperscriptBox["z", "2"], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["b", "3"], " ", SqrtBox["\[Pi]"]]]]]]]]

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

 2001-10-29