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http://functions.wolfram.com/06.33.21.0046.01
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Integrate[(Cos[b Sqrt[z]] FresnelC[a Sqrt[z]])/Sqrt[z], z] ==
(1/b) ((-Cos[b^2/(2 a^2 Pi)]) FresnelS[b/(a Pi) - a Sqrt[z]] -
Cos[b^2/(2 a^2 Pi)] FresnelS[b/(a Pi) + a Sqrt[z]] +
FresnelC[b/(a Pi) - a Sqrt[z]] Sin[b^2/(2 a^2 Pi)] +
FresnelC[b/(a Pi) + a Sqrt[z]] Sin[b^2/(2 a^2 Pi)] +
2 FresnelC[a Sqrt[z]] Sin[b Sqrt[z]])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]]]], SqrtBox["z"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "b"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "]"]]]], " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox["b", RowBox[List["a", " ", "\[Pi]"]]], "-", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Cos", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "]"]], " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox["b", RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox["b", RowBox[List["a", " ", "\[Pi]"]]], "-", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox["b", RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]]]], ")"]]]]]]]]
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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> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </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> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> b </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mfrac> <mi> b </mi> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mi> b </mi> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mfrac> <mi> b </mi> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mi> b </mi> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </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> <cos /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> FresnelS </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cos /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> FresnelS </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> FresnelC </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> FresnelC </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </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[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", SqrtBox["z_"]]], "]"]]]], SqrtBox["z_"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Cos", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "]"]]]], " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox["b", RowBox[List["a", " ", "\[Pi]"]]], "-", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]]]], "-", RowBox[List[RowBox[List["Cos", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "]"]], " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox["b", RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox["b", RowBox[List["a", " ", "\[Pi]"]]], "-", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox["b", RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[SuperscriptBox["b", "2"], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]]]], "b"]]]]] |
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Date Added to functions.wolfram.com (modification date)
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