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; }

 Erf

 http://functions.wolfram.com/06.25.21.0109.01

 Input Form

 Integrate[z E^(b z) Cosh[c z] Erf[a z], z] == (1/(4 a^2 Sqrt[Pi])) (((1/(b - c)^2) (2 a (b - c) E^((b - c) z) + 2 a^2 E^(z (b - c + a^2 z)) Sqrt[Pi] (-1 + b z - c z) Erf[a z] - (2 a^2 - (b - c)^2) E^((b - c)^2/(4 a^2) + a^2 z^2) Sqrt[Pi] Erf[(b - c)/(2 a) - a z]) + (1/(b + c)^2) (2 a (b + c) E^((b + c) z) + 2 a^2 E^(z (b + c + a^2 z)) Sqrt[Pi] (-1 + b z + c z) Erf[a z] - (2 a^2 - (b + c)^2) 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["z", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox["a", "2"], " ", SqrtBox["\[Pi]"]]]], RowBox[List["(", RowBox[List[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"]], ")"]], "2"]], 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["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "-", "c", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["b", " ", "z"]], "-", RowBox[List["c", " ", "z"]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], ")"]], " ", 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"]], ")"]], "2"]], 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["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List["c", " ", "z"]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"]]], ")"]], " ", 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 b z cosh ( c z ) erf ( a z ) z 1 4 a 2 π - a 2 z 2 ( 1 ( b - c ) 2 ( 2 z ( z a 2 + b - c ) π ( b z - c z - 1 ) erf ( a z ) a 2 + 2 ( b - c ) ( b - c ) z a - ( 2 a 2 - ( b - c ) 2 ) ( b - c ) 2 4 a 2 + a 2 z 2 π erf ( b - c 2 a - a z ) ) + 1 ( b + c ) 2 ( 2 z ( z a 2 + b + c ) π ( b z + c z - 1 ) erf ( a z ) a 2 + 2 ( b + c ) ( b + c ) z a - ( 2 a 2 - ( b + c ) 2 ) ( b + c ) 2 4 a 2 + a 2 z 2 π erf ( b + c 2 a - a z ) ) ) z z b z c z Erf a z 1 4 a 2 1 2 -1 -1 a 2 z 2 1 b -1 c 2 -1 2 z z a 2 b -1 c 1 2 b z -1 c z -1 Erf a z a 2 2 b -1 c b -1 c z a -1 2 a 2 -1 b -1 c 2 b -1 c 2 4 a 2 -1 a 2 z 2 1 2 Erf b -1 c 2 a -1 -1 a z 1 b c 2 -1 2 z z a 2 b c 1 2 b z c z -1 Erf a z a 2 2 b c b c z a -1 2 a 2 -1 b c 2 b c 2 4 a 2 -1 a 2 z 2 1 2 Erf b c 2 a -1 -1 a z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["z_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Erf", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", 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["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "-", "c", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["b", " ", "z"]], "-", RowBox[List["c", " ", "z"]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], ")"]], " ", 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"]], ")"]], "2"]], "+", 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["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "+", "c", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List["c", " ", "z"]]]], ")"]], " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"]]], ")"]], " ", 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"]], ")"]], "2"]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["a", "2"], " ", SqrtBox["\[Pi]"]]]]]]]]

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

 2001-10-29