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 Erfi

 http://functions.wolfram.com/06.28.21.0138.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29