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http://functions.wolfram.com/01.07.21.0470.01
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Integrate[Cos[a z^2 + b/z^2], z] == (1/8) Sqrt[Pi]
((1/Sqrt[(-I) a]) ((1 - Erf[Sqrt[(-I) b]/z - Sqrt[(-I) a] z] +
E^(4 Sqrt[(-I) a] Sqrt[(-I) b])
(-1 + Erf[Sqrt[(-I) b]/z + Sqrt[(-I) a] z]))/
E^(2 Sqrt[(-I) a] Sqrt[(-I) b])) - (1/Sqrt[I a])
((-1 + Erf[Sqrt[I b]/z - Sqrt[I a] z] - E^(4 Sqrt[I a] Sqrt[I b])
(-1 + Erf[Sqrt[I b]/z + Sqrt[I a] z]))/E^(2 Sqrt[I a] Sqrt[I b])))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", FractionBox["b", SuperscriptBox["z", "2"]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "8"], SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]]]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Erf", "[", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]], "z"], "-", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], " ", "z"]]]], "]"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Erf", "[", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]], "z"], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a"]]], " ", "z"]]]], "]"]]]], ")"]]]]]], ")"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "a"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "a"]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Erf", "[", RowBox[List[FractionBox[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]], "z"], "-", RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "a"]]], " ", "z"]]]], "]"]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "a"]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Erf", "[", RowBox[List[FractionBox[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]], "z"], "+", RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "a"]]], " ", "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> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mfrac> <mi> b </mi> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> <mi> z </mi> </mfrac> <mo> - </mo> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> <mi> z </mi> </mfrac> <mo> - </mo> <mrow> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </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> <cos /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> Erf </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Erf </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> Erf </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Erf </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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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