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http://functions.wolfram.com/06.33.21.0076.01
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Integrate[z^3 Sinh[b z^2] FresnelC[a z], z] ==
(I/(16 b^2)) (-8 I FresnelC[a z] (b z^2 Cosh[b z^2] - Sinh[b z^2]) +
a ((4 Sqrt[Pi] (-3 I b + a^2 Pi) FresnelS[Sqrt[a^2 - (2 I b)/Pi] z])/
(-2 I b + a^2 Pi)^(3/2) - (4 Sqrt[Pi] (3 I b + a^2 Pi)
FresnelS[Sqrt[a^2 + (2 I b)/Pi] z])/(2 I b + a^2 Pi)^(3/2) +
((I b)/z) (Sqrt[2 Pi] (Sqrt[(2 b - I a^2 Pi) z^2]/(2 I b + a^2 Pi)^2 +
Sqrt[(-(2 b + I a^2 Pi)) z^2]/(-2 I b + a^2 Pi)^2 -
Sqrt[(2 b + I a^2 Pi) z^2]/(2 b + I a^2 Pi)^2 +
Sqrt[I (2 I b + a^2 Pi) z^2]/(2 I b + a^2 Pi)^2) +
((8 z^2)/(4 b^2 + a^4 Pi^2)) (a^2 Pi Cosh[b z^2]
Sin[(1/2) a^2 Pi z^2] + 2 b Cos[(1/2) a^2 Pi z^2] Sinh[b z^2]))))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List["16", " ", SuperscriptBox["b", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "\[ImaginaryI]", " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], ")"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "\[Pi]"]]]], " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["4", " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "\[Pi]"]]]], " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], "z"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], "2"]], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], "2"]], "-", FractionBox[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], "2"]], "+", FractionBox[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], "2"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["8", " ", SuperscriptBox["z", "2"], " "]], RowBox[List[RowBox[List["4", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", 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> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </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> <mi> ⅈ </mi> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mi> π </mi> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <msqrt> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mi> π </mi> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <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> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <msqrt> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msup> <mrow> <mo> ( </mo> <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> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <msqrt> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mrow> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </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> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> cos </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> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <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> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </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> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <sinh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> <imaginaryi /> </apply> </apply> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <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> <ci> b </ci> <imaginaryi /> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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'> 3 </cn> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> b </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <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 /> <ci> b </ci> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <imaginaryi /> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> 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<plus /> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <cosh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </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> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <cos /> <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> <sinh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <imaginaryi /> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <cosh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sinh /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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