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/06.28.21.0049.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29