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http://functions.wolfram.com/06.28.21.0039.01
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Integrate[z^3 Sin[b z^2] Erfi[a z], z] ==
(1/(4 b^2))
((1/Sqrt[Pi]) (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/(a^4 + b^2))
(a (Sqrt[a^2 - I b] ((-I) a^2 + b) Erfi[Sqrt[a^2 - I b] z] +
Sqrt[a^2 + I b] (I a^2 + b) Erfi[Sqrt[a^2 + I b] z])) -
2 Erfi[a z] (b z^2 Cos[b z^2] - Sin[b z^2]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox["b", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SqrtBox["\[Pi]"]], RowBox[List["(", RowBox[List["a", " ", "b", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", 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[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "2"]]]], RowBox[List["(", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", "z"]], "]"]]]]]], ")"]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> π </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> 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<cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> b </ci> <imaginaryi /> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> b </ci> <imaginaryi /> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Erfi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
