Fresnel integral S: Representations through more general functions (formula 06.32.26.0009)

FresnelS

$Mathematica Notation$

Gamma, Beta, Erf

FresnelS[z]

Representations through more general functions

Through Meijer G

Generalized cases involving cos,sin and Fresnel C

http://functions.wolfram.com/06.32.26.0009.01

Input Form

Cos[(Pi z^2)/2] FresnelC[z] + Sin[(Pi z^2)/2] FresnelS[z] == Sqrt[Pi/2] MeijerG[{{1/4}, {}}, {{1/4}, {0, 1/2}}, (Sqrt[Pi] z)/2, 1/4]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z", "2"]]], "2"], "]"]], " ", RowBox[List["FresnelC", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z", "2"]]], "2"], "]"]], " ", RowBox[List["FresnelS", "[", "z", "]"]]]]]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "4"], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "4"], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", "z"]], "2"], ",", FractionBox["1", "4"]]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mtext> </mtext> <mrow> <mrow> <mi> sin </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 3 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["1", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", "z"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True]], ",", TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> cos </ci> <apply> <times /> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> FresnelC </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <ci> sin </ci> <apply> <times /> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> FresnelS </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> MeijerG </ci> <list> <list> <cn type='rational'> 1 <sep /> 4 </cn> </list> <list /> </list> <list> <list> <cn type='rational'> 1 <sep /> 4 </cn> </list> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </list> </list> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z_", "2"]]], "2"], "]"]], " ", RowBox[List["FresnelC", "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z_", "2"]]], "2"], "]"]], " ", RowBox[List["FresnelS", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "4"], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "4"], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", "z"]], "2"], ",", FractionBox["1", "4"]]], "]"]]]]]]]]

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

2001-10-29