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http://functions.wolfram.com/06.33.21.0120.01
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Integrate[Erfc[b z] FresnelC[a z], z] ==
(1/(4 a b Pi)) ((1 - I) Sqrt[-2 I b^2 + a^2 Pi]
Erf[((1 + I)/2) Sqrt[-2 I b^2 + a^2 Pi] z] +
(1 - I) Sqrt[2 I b^2 + a^2 Pi] Erfi[((1 + I)/2) Sqrt[2 I b^2 + a^2 Pi]
z] - (4 a Sqrt[Pi] FresnelC[a z])/E^(b^2 z^2) +
4 a b Pi z Erfc[b z] FresnelC[a z] - 4 b Erfc[b z] Sin[(1/2) a^2 Pi z^2])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", "a", " ", "b", " ", "\[Pi]"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"], SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"], SqrtBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], "-", RowBox[List["4", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["4", " ", "a", " ", "b", " ", "\[Pi]", " ", "z", " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List["4", " ", "b", " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", 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> <mrow> <mrow> <mi> erfc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </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> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> erfc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> erfc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <ci> Erfc </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> b </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='complex-cartesian'> 1 <sep /> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erf </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='complex-cartesian'> 1 <sep /> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <ci> b </ci> <pi /> <ci> z </ci> <apply> <ci> Erfc </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> <apply> <ci> Erfc </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </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["Erfc", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", RowBox[List["Erf", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", RowBox[List["Erfi", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], "-", RowBox[List["4", " ", "a", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", SuperscriptBox["b", "2"]]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List["4", " ", "a", " ", "b", " ", "\[Pi]", " ", "z", " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "-", RowBox[List["4", " ", "b", " ", RowBox[List["Erfc", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], RowBox[List["4", " ", "a", " ", "b", " ", "\[Pi]"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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