Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











FresnelC






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelC[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving power, exponential and hyperbolic functions > Involving power, exp and cosh





http://functions.wolfram.com/06.33.21.0100.01









  


  










Input Form





Integrate[z^3 E^(b z) Cosh[c z] FresnelC[a z], z] == ((1/(4 a^6 Pi^3)) (((1/(b - c)^4) (a (b - c) E^((I (b - c)^2)/(2 a^2 Pi) + (b - c) z) (I (b - c)^4 + a^2 (b - c)^2 Pi - 6 I a^4 Pi^2 - a^2 (b - c)^3 Pi z + 3 I a^4 (b - c) Pi^2 z - I a^4 (b - c)^2 Pi^2 z^2 + E^(I a^2 Pi z^2) ((-I) (b - c)^4 + 6 I a^4 Pi^2 - a^2 (b - c)^3 Pi z - 3 I a^4 (b - c) Pi^2 z + a^2 (b - c)^2 Pi (1 + I a^2 Pi z^2))) + E^((1/2) I a^2 Pi z^2) (2 a^6 E^((I (b - c)^2)/(2 a^2 Pi) + (b - c) z) Pi^3 (-6 + (b - c) z (6 + (b - c) z (-3 + b z - c z))) FresnelC[a z] + E^((I (b - c)^2)/(a^2 Pi)) ((b - c)^6 - 3 a^4 (b - c)^2 Pi^2 - 6 I a^6 Pi^3) (FresnelC[(b - c)/(a Pi) + I a z] - I FresnelS[(b - c)/(a Pi) + I a z]) + ((-I) (b - c)^6 + 3 I a^4 (b - c)^2 Pi^2 + 6 a^6 Pi^3) (FresnelC[(I (b - c))/(a Pi) + a z] - I FresnelS[(I (b - c))/(a Pi) + a z]))))/ E^((I (b - c)^2)/(2 a^2 Pi)) + ((1/(b + c)^4) (a (b + c) E^(((b + c) (I b + I c + 2 a^2 Pi z))/ (2 a^2 Pi)) (I (b + c)^4 + a^2 (b + c)^2 Pi - 6 I a^4 Pi^2 - a^2 (b + c)^3 Pi z + 3 I a^4 (b + c) Pi^2 z - I a^4 (b + c)^2 Pi^2 z^2 + E^(I a^2 Pi z^2) ((-I) (b + c)^4 + 6 I a^4 Pi^2 - a^2 (b + c)^3 Pi z - 3 I a^4 (b + c) Pi^2 z + a^2 (b + c)^2 Pi (1 + I a^2 Pi z^2))) + E^((1/2) I a^2 Pi z^2) (2 a^6 E^(((b + c) (I b + I c + 2 a^2 Pi z))/(2 a^2 Pi)) Pi^3 (-6 + (b + c) z (6 + (b + c) z (-3 + b z + c z))) FresnelC[a z] + E^((I (b + c)^2)/(a^2 Pi)) ((b + c)^6 - 3 a^4 (b + c)^2 Pi^2 - 6 I a^6 Pi^3) (FresnelC[(b + c)/(a Pi) + I a z] - I FresnelS[(b + c)/(a Pi) + I a z]) + ((-I) (b + c)^6 + 3 I a^4 (b + c)^2 Pi^2 + 6 a^6 Pi^3) (FresnelC[(I b + I c + a^2 Pi z)/(a Pi)] - I FresnelS[(I b + I c + a^2 Pi z)/(a Pi)]))))/ E^((I (b + c)^2)/(2 a^2 Pi))))/E^((1/2) I a^2 Pi z^2)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", "z"]]], RowBox[List["Cosh", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "4"]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "4"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", "\[Pi]"]], "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "3"], " ", "\[Pi]", " ", "z"]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "4"]]], "+", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "3"], " ", "\[Pi]", " ", "z"]], "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z"]]]]], " ", SuperscriptBox["\[Pi]", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["b", " ", "z"]], "-", RowBox[List["c", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "6"], "-", RowBox[List["3", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "6"]]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["6", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "4"]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "4"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", "\[Pi]"]], "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "3"], " ", "\[Pi]", " ", "z"]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "4"]]], "+", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "3"], " ", "\[Pi]", " ", "z"]], "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", SuperscriptBox["\[Pi]", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"]]], RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "6"], "-", RowBox[List["3", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "6"]]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["6", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", "z"]]]], RowBox[List["a", " ", "\[Pi]"]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", "z"]]]], RowBox[List["a", " ", "\[Pi]"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox[&quot;C&quot;, FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </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> <power /> <exponentiale /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <cosh /> <apply> <times /> <ci> c </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> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <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> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <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> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <imaginaryi /> </apply> <apply> <times /> <ci> c </ci> <imaginaryi /> </apply> </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> <plus /> <apply> <times /> <cn type='integer'> -6 </cn> <imaginaryi /> <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'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 3 </cn> </apply> <pi /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 4 </cn> </apply> <imaginaryi /> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <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> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <imaginaryi /> <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'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 3 </cn> </apply> <pi /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <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> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <imaginaryi /> </apply> <apply> <times /> <ci> c </ci> <imaginaryi /> </apply> </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> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -6 </cn> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <imaginaryi /> </apply> <apply> <times /> <ci> c </ci> <imaginaryi /> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <pi /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <imaginaryi /> </apply> <apply> <times /> <ci> c </ci> <imaginaryi /> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -6 </cn> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> FresnelC </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> FresnelS </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <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> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <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> <times /> <ci> z </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -6 </cn> <imaginaryi /> <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'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <pi /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <imaginaryi /> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <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> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <imaginaryi /> <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'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <pi /> <ci> z </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <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> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <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> <times /> <ci> z </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -6 </cn> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> FresnelC </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> FresnelS </ci> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -6 </cn> <imaginaryi /> <apply> <power /> <pi /> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> FresnelC </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> FresnelS </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", "z_"]]], " ", RowBox[List["Cosh", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "4"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", "\[Pi]"]], "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "3"], " ", "\[Pi]", " ", "z"]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "4"]]], "+", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "3"], " ", "\[Pi]", " ", "z"]], "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z"]]]]], " ", SuperscriptBox["\[Pi]", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["b", " ", "z"]], "-", RowBox[List["c", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"]]], RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "6"], "-", RowBox[List["3", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox[RowBox[List["b", "-", "c"]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "6"]]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["6", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]]]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], "4"]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "4"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", "\[Pi]"]], "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "3"], " ", "\[Pi]", " ", "z"]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "4"]]], "+", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "3"], " ", "\[Pi]", " ", "z"]], "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", SuperscriptBox["\[Pi]", "2"], " ", "z"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", "z"]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", SuperscriptBox["\[Pi]", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["b", " ", "z"]], "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"]]], RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "6"], "-", RowBox[List["3", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", RowBox[List[FractionBox[RowBox[List["b", "+", "c"]], RowBox[List["a", " ", "\[Pi]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", "z"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "6"]]], "+", RowBox[List["3", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["6", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", "z"]]]], RowBox[List["a", " ", "\[Pi]"]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", "z"]]]], RowBox[List["a", " ", "\[Pi]"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", "c"]], ")"]], "4"]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["a", "6"], " ", SuperscriptBox["\[Pi]", "3"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29