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FresnelS




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelS[z] > Integral representations > Contour integral representations





http://functions.wolfram.com/06.32.07.0004.01









  


  








Input Form





FresnelS[z] == ((Pi/Sqrt[2]) (1/(2 Pi I)) ContourIntegrate[ Gamma[s + 3/4]/(Gamma[s + 1] Gamma[3/4 - s] Gamma[1 - s])/ ((1/2) E^(Pi (I/4)) Sqrt[Pi] z)^(4 s), {s, \[ScriptCapitalL]}])/ E^(3 Pi (I/4))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["FresnelS", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", SqrtBox["2"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "3"]], " ", "\[Pi]", " ", RowBox[List["\[ImaginaryI]", "/", "4"]]]]], FractionBox[RowBox[List["1", " "]], RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["3", "4"]]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "4"], "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", RowBox[List["\[ImaginaryI]", "/", "4"]]]]], " ", SqrtBox["\[Pi]"], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "4"]], "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]]]]]]








MathML Form







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</mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> &#8290; </mo> <mi> s </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> s </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> S </ms> <ci> FresnelC </ci> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; 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</ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> s </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 3 </ms> <ms> 4 </ms> </apply> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> &#8519; </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <ms> &#8520; </ms> </list> </apply> <ms> 4 </ms> </apply> </apply> <apply> <ci> SqrtBox </ci> <ms> &#960; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FresnelS", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "4"]]]], " ", RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["3", "4"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "4"]], " ", SqrtBox["\[Pi]"], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "4"]], " ", "s"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "4"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]]]]


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





2001-10-29