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; }

 Erfi

 http://functions.wolfram.com/06.28.21.0066.01

 Input Form

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

 Standard Form

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

 MathML Form

 z 2 b z cos ( c z ) erfi ( a z ) 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 ) erfi ( a z ) a 4 + 2 ( b + c ) ( b + c ) z ( ( b + c ) 2 - 2 a 2 ( b z + c z - 2 ) ) a - ( 8 a 4 + 2 ( b + c ) 2 a 2 + ( b + c ) 4 ) - ( b + c ) 2 4 a 2 - a 2 z 2 π erfi ( 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 ) erfi ( a z ) a 4 + 2 ( b - c ) ( b - c ) z ( ( - 2 b z + 2 c z + 4 ) a 2 + ( b - c ) 2 ) a - ( 8 a 4 + 2 ( b - c ) 2 a 2 + ( b - c ) 4 ) - ( b - c ) 2 4 a 2 - a 2 z 2 π erfi ( b - c 2 a + a z ) ) ) ) z z 2 b z c z Erfi a z 1 8 a 4 1 2 -1 a 2 z 2 1 b c 3 -1 4 z -1 z a 2 b c 1 2 b c 2 z 2 -1 2 b c z 2 Erfi a z a 4 2 b c b c z b c 2 -1 2 a 2 b z c z -2 a -1 8 a 4 2 b c 2 a 2 b c 4 -1 b c 2 4 a 2 -1 -1 a 2 z 2 1 2 Erfi b c 2 a -1 a z 1 b -1 c 3 -1 4 z -1 z a 2 b -1 c 1 2 b -1 c 2 z 2 -1 2 b -1 c z 2 Erfi a z a 4 2 b -1 c b -1 c z -2 b z 2 c z 4 a 2 b -1 c 2 a -1 8 a 4 2 b -1 c 2 a 2 b -1 c 4 -1 b -1 c 2 4 a 2 -1 -1 a 2 z 2 1 2 Erfi b -1 c 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"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Erfi", "[", 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", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], RowBox[List["2", " ", "a"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "3"]], "+", FractionBox[RowBox[List[RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], "-", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "z"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "z"]]]], ")"]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", "z"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["Erfi", "[", 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", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", SuperscriptBox["a", "2"]]]]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], RowBox[List["2", " ", "a"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "3"]]]], ")"]]]], RowBox[List["8", " ", SuperscriptBox["a", "4"], " ", SqrtBox["\[Pi]"]]]]]]]]

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

 2001-10-29