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FresnelS






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelS[z] > Representations through more general functions > Through hypergeometric functions > Involving hypergeometric U





http://functions.wolfram.com/06.32.26.0002.01









  


  










Input Form





FresnelS[z] == ((I z)/(2 Sqrt[2])) ((1/Sqrt[I z^2]) (1 - ((1/Sqrt[Pi]) HypergeometricU[1/2, 1/2, ((I Pi)/2) z^2])/E^(((I Pi)/2) z^2)) - (1/Sqrt[(-I) z^2]) (1 - (1/Sqrt[Pi]) E^(((I Pi)/2) z^2) HypergeometricU[1/2, 1/2, (-((I Pi)/2)) z^2]))










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> S </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox[&quot;U&quot;, HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FresnelS", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["1", "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], SqrtBox["\[Pi]"]]]], SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]], "-", FractionBox[RowBox[List["1", "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["-", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]], "]"]]]], SqrtBox["\[Pi]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["z", "2"]]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["2"]]]]]]]]










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





2001-10-29





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