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Erfi






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erfi[z] > Integration > Indefinite integration > Involving direct function and Gamma-, Beta-, Erf-type functions > Involving erf-type functions and a power function > Involving erf and power





http://functions.wolfram.com/06.28.21.0135.01









  


  










Input Form





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










Standard Form





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MathML Form







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1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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