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FresnelC






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.33.07.0004.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["FresnelC", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " "]], SqrtBox["2"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["\[Pi]", "/", "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["1", "4"]]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "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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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FresnelC", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], " ", "\[Pi]"]]], " ", RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "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["1", "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





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