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http://functions.wolfram.com/06.28.21.0121.01
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Integrate[Erfi[a z] Erfi[b z], z] ==
-((E^(b^2 z^2) Erfi[a z])/(b Sqrt[Pi])) +
((-E^(a^2 z^2) + a Sqrt[Pi] z Erfi[a z]) Erfi[b z])/(a Sqrt[Pi]) +
(a Erfi[(a^2 z + b^2 z)/Sqrt[a^2 + b^2]])/(b Sqrt[a^2 + b^2] Sqrt[Pi]) +
(b Erfi[(a^2 z + b^2 z)/Sqrt[a^2 + b^2]])/(a Sqrt[a^2 + b^2] Sqrt[Pi])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], RowBox[List["b", " ", SqrtBox["\[Pi]"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List["a", " ", SqrtBox["\[Pi]"], " ", "z", " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List["b", " ", "z"]], "]"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", "z"]]]], SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], RowBox[List["b", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]], " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List["b", " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", "z"]]]], SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], RowBox[List["a", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]], " ", SqrtBox["\[Pi]"]]]]]]]]]]
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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> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </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> <mrow> <mfrac> <mrow> <mi> b </mi> <mtext> </mtext> </mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mtext> </mtext> </mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <ci> Erfi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <ci> Erfi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Erfi", "[", RowBox[List["a_", " ", "z_"]], "]"]], " ", RowBox[List["Erfi", "[", RowBox[List["b_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["b", "2"], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]], RowBox[List["b", " ", SqrtBox["\[Pi]"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List["a", " ", SqrtBox["\[Pi]"], " ", "z", " ", RowBox[List["Erfi", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List["b", " ", "z"]], "]"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", "z"]]]], SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], RowBox[List["b", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]], " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List["b", " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["b", "2"], " ", "z"]]]], SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]]], "]"]]]], RowBox[List["a", " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"]]]], " ", SqrtBox["\[Pi]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
