Fresnel integral S: Representations through equivalent functions (formula 06.32.27.0001)

FresnelS

$Mathematica Notation$

Gamma, Beta, Erf

FresnelS[z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/06.32.27.0001.01

Input Form

FresnelS[z] == ((1 + I)/4) (Erf[((1 + I)/2) Sqrt[Pi] z] - I Erf[((1 - I)/2) Sqrt[Pi] z])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["FresnelS", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["Erf", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "\[ImaginaryI]"]], "2"], " ", SqrtBox["\[Pi]"], " ", "z"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Erf", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[ImaginaryI]"]], "2"], SqrtBox["\[Pi]"], " ", "z"]], "]"]]]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> FresnelS </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Erf </ci> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> Erf </ci> <apply> <times /> <apply> <times /> <cn type='complex-cartesian'> 1 <sep /> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FresnelS", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Erf", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SqrtBox["\[Pi]"], " ", "z"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Erf", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SqrtBox["\[Pi]"], " ", "z"]], "]"]]]]]], ")"]]]]]]]]

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

2001-10-29