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 Erfi

 http://functions.wolfram.com/06.28.21.0127.01

 Input Form

 Integrate[z^2 Erfi[a z] Erfi[b 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]) - (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 Erfi[b z])/a^3 - (E^(a^2 z^2) Pi z^2 Erfi[b z])/a + (1/b^3) (Pi Erfi[a z] (E^(b^2 z^2) (1 - b^2 z^2) + b^3 Sqrt[Pi] z^3 Erfi[b z])) - (a Pi Erfi[Sqrt[a^2 + b^2] z])/ (b^3 Sqrt[a^2 + b^2]) - (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["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["Erfi", "[", RowBox[List["b", " ", "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["\[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["Erfi", "[", 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["Erfi", "[", RowBox[List["b", " ", "z"]], "]"]]]], "a"], "+", RowBox[List[FractionBox["1", SuperscriptBox["b", "3"]], RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["b", "3"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Erfi", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], ")"]]]], "-", FractionBox[RowBox[List["a", " ", "\[Pi]", " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]], " ", "z"]], "]"]]]], RowBox[List[SuperscriptBox["b", "3"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]]]], "-", 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 erfi ( a z ) erfi ( b z ) z 1 3 π 3 / 2 ( - a 2 z 2 π erfi ( b z ) z 2 a + ( a 2 + b 2 ) z 2 π z a b - π erf ( - ( a 2 + b 2 ) z 2 ) z 2 a b - ( a 2 + b 2 ) z 2 + π z 2 a b - ( a 2 + b 2 ) z 2 + a 2 z 2 π erfi ( b z ) a 3 + π erfi ( a z ) ( b 3 π erfi ( b z ) z 3 + b 2 z 2 ( 1 - b 2 z 2 ) ) b 3 - b π erfi ( a 2 + b 2 z ) a 3 a 2 + b 2 - a π erfi ( a 2 + b 2 z ) b 3 a 2 + b 2 ) z z 2 Erfi a z Erfi b z 1 3 3 2 -1 -1 a 2 z 2 Erfi b z z 2 a -1 a 2 b 2 z 2 1 2 z a b -1 -1 Erf -1 a 2 b 2 z 2 1 2 z 2 a b -1 a 2 b 2 z 2 1 2 -1 z 2 a b -1 a 2 b 2 z 2 1 2 -1 a 2 z 2 Erfi b z a 3 -1 Erfi a z b 3 1 2 Erfi b z z 3 b 2 z 2 1 -1 b 2 z 2 b 3 -1 -1 b Erfi a 2 b 2 1 2 z a 3 a 2 b 2 1 2 -1 -1 a Erfi a 2 b 2 1 2 z b 3 a 2 b 2 1 2 -1 [/itex]

 Rule Form

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

 2001-10-29