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http://functions.wolfram.com/01.20.21.0722.01
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Integrate[Cos[b z^2 + e] Cosh[c z^2], z] ==
(1/2) Sqrt[Pi/2] ((1/Sqrt[-b + I c])
(Cos[e] FresnelC[Sqrt[-b + I c] Sqrt[2/Pi] z] +
FresnelS[Sqrt[-b + I c] Sqrt[2/Pi] z] Sin[e]) +
(1/Sqrt[-b - I c]) (Cos[e] FresnelC[Sqrt[-b - I c] Sqrt[2/Pi] z] +
FresnelS[Sqrt[-b - I c] Sqrt[2/Pi] z] Sin[e]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "+", "e"]], "]"]], RowBox[List["Cosh", "[", RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "e", "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", "e", "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "e", "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", "e", "]"]]]]]], ")"]]]]]], ")"]]]]]]]]
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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> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> e </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> e </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mrow> </msqrt> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> e </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> e </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <mi> b </mi> </mrow> </msqrt> </mfrac> </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> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> e </ci> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <ci> e </ci> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <sin /> <ci> e </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cos /> <ci> e </ci> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <sin /> <ci> e </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </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["Cos", "[", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", "e_"]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["Cos", "[", "e", "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", "e", "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["Cos", "[", "e", "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", "e", "]"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]]]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
