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/06.27.21.0109.01

 Input Form

 Integrate[z^2 E^(b z) Cosh[c z] Erfc[a z], z] == (E^(b z)/(b^2 - c^2)^3) ((-2 b^4 z + 2 c^4 z + b^5 z^2 + b^3 (2 - 2 c^2 z^2) + b c^2 (6 + c^2 z^2)) Cosh[c z] - c (-4 b^3 z + 4 b c^2 z + b^4 z^2 + b^2 (6 - 2 c^2 z^2) + c^2 (2 + c^2 z^2)) Sinh[c z]) - ((1/(8 a^4 Sqrt[Pi])) ((1/(b + c)^3) (2 a (b + c) E^((b + c) z) ((b + c)^2 + 2 a^2 (-2 + b z + c z)) + 4 a^4 E^(z (b + c + a^2 z)) Sqrt[Pi] (2 - 2 (b + c) z + (b + c)^2 z^2) Erf[a z] + (8 a^4 - 2 a^2 (b + c)^2 + (b + c)^4) E^((b + c)^2/(4 a^2) + a^2 z^2) Sqrt[Pi] Erf[(b + c)/(2 a) - a z]) + (1/(b - c)^3) (2 a (b - c) E^((b - c) z) ((b - c)^2 + 2 a^2 (-2 + b z - c z)) + 4 a^4 E^(z (b - c + a^2 z)) Sqrt[Pi] (2 - 2 (b - c) z + (b - c)^2 z^2) Erf[a z] + (8 a^4 - 2 a^2 (b - c)^2 + (b - c)^4) E^((b - c)^2/(4 a^2) + a^2 z^2) Sqrt[Pi] Erf[(b - c)/(2 a) - a z])))/ E^(a^2 z^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], "3"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["b", "4"], " ", "z"]], "+", RowBox[List["2", " ", SuperscriptBox["c", "4"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "5"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List["b", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["b", "3"], " ", "z"]], "+", RowBox[List["4", " ", "b", " ", SuperscriptBox["c", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["6", "-", RowBox[List["2", " ", SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SqrtBox["\[Pi]"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "3"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", "a"]]], "-", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "3"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["b", " ", "z"]], "-", RowBox[List["c", " ", "z"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "-", "c", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", "a"]]], "-", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]]

 MathML Form

 z 2 b z cosh ( c z ) erfc ( a z ) z b z ( b 2 - c 2 ) 3 ( ( z 2 b 5 - 2 z b 4 + ( 2 - 2 c 2 z 2 ) b 3 + c 2 ( c 2 z 2 + 6 ) b + 2 c 4 z ) cosh ( c z ) - c ( z 2 b 4 - 4 z b 3 + ( 6 - 2 c 2 z 2 ) b 2 + 4 c 2 z b + c 2 ( c 2 z 2 + 2 ) ) sinh ( c z ) ) - 1 8 a 4 π - a 2 z 2 ( 1 ( b + c ) 3 ( 4 z ( z a 2 + b + c ) π ( ( b + c ) 2 z 2 - 2 ( b + c ) z + 2 ) erf ( a z ) a 4 + 2 ( b + c ) ( b + c ) z ( 2 ( b z + c z - 2 ) a 2 + ( b + c ) 2 ) a + ( 8 a 4 - 2 ( b + c ) 2 a 2 + ( b + c ) 4 ) exp ( ( b + c ) 2 4 a 2 + a 2 z 2 ) π erf ( b + c 2 a - a z ) ) + 1 ( b - c ) 3 ( 4 z ( z a 2 + b - c ) π ( ( b - c ) 2 z 2 - 2 ( b - c ) z + 2 ) erf ( a z ) a 4 + 2 ( b - c ) ( b - c ) z ( 2 ( b z - c z - 2 ) a 2 + ( b - c ) 2 ) a + ( 8 a 4 - 2 ( b - c ) 2 a 2 + ( b - c ) 4 ) exp ( ( b - c ) 2 4 a 2 + a 2 z 2 ) π erf ( b - c 2 a - a z ) ) ) z z 2 b z c z Erfc a z b z b 2 -1 c 2 3 -1 z 2 b 5 -1 2 z b 4 2 -1 2 c 2 z 2 b 3 c 2 c 2 z 2 6 b 2 c 4 z c z -1 c z 2 b 4 -1 4 z b 3 6 -1 2 c 2 z 2 b 2 4 c 2 z b c 2 c 2 z 2 2 c z -1 1 8 a 4 1 2 -1 -1 a 2 z 2 1 b c 3 -1 4 z z a 2 b c 1 2 b c 2 z 2 -1 2 b c z 2 Erf a z a 4 2 b c b c z 2 b z c z -2 a 2 b c 2 a 8 a 4 -1 2 b c 2 a 2 b c 4 b c 2 4 a 2 -1 a 2 z 2 1 2 Erf b c 2 a -1 -1 a z 1 b -1 c 3 -1 4 z z a 2 b -1 c 1 2 b -1 c 2 z 2 -1 2 b -1 c z 2 Erf a z a 4 2 b -1 c b -1 c z 2 b z -1 c z -2 a 2 b -1 c 2 a 8 a 4 -1 2 b -1 c 2 a 2 b -1 c 4 b -1 c 2 4 a 2 -1 a 2 z 2 1 2 Erf b -1 c 2 a -1 -1 a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Erfc", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["b", "4"], " ", "z"]], "+", RowBox[List["2", " ", SuperscriptBox["c", "4"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "5"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List["b", " ", SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]]]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["b", "3"], " ", "z"]], "+", RowBox[List["4", " ", "b", " ", SuperscriptBox["c", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "4"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["6", "-", RowBox[List["2", " ", SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["c", "2"], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["c", " ", "z"]], "]"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["b", "2"], "-", SuperscriptBox["c", "2"]]], ")"]], "3"]], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["2", " ", "a"]]], "-", RowBox[List["a", " ", "z"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "3"]], "+", FractionBox[RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["b", " ", "z"]], "-", RowBox[List["c", " ", "z"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "-", "c", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["8", " ", SuperscriptBox["a", "4"]]], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["2", " ", "a"]]], "-", RowBox[List["a", " ", "z"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "3"]]]], ")"]]]], RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SqrtBox["\[Pi]"]]]]]]]]]]

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

 2001-10-29