Fresnel integral S: Integration (formula 06.32.21.0005)

FresnelS

$Mathematica Notation$

Gamma, Beta, Erf

FresnelS[z]

Integration

Indefinite integration

Involving one direct function and a power function

Linear arguments

http://functions.wolfram.com/06.32.21.0005.01

Input Form

Integrate[z^n FresnelS[a z], z] == (1/(2 a (1 + n))) ((Pi^(-1 - n/2) z^n (2 a Pi^((2 + n)/2) z (a^4 z^4)^(n/2) FresnelS[a z] + 2^(n/2) ((I a^2 z^2)^(n/2) Gamma[(2 + n)/2, (-(1/2)) I a^2 Pi z^2] + ((-I) a^2 z^2)^(n/2) Gamma[(2 + n)/2, (1/2) I a^2 Pi z^2])))/ (a^4 z^4)^(n/2))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["FresnelS", "[", RowBox[List["a", " ", "z"]], "]"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["n", "2"]]]], " ", SuperscriptBox["z", "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List[RowBox[List["-", "n"]], "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", SuperscriptBox["\[Pi]", FractionBox[RowBox[List["2", "+", "n"]], "2"]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List["n", "/", "2"]]], " ", RowBox[List["FresnelS", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["n", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["n", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["n", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "n"]], "2"], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], ")"]]]]]]]]

MathML Form

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

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", RowBox[List["FresnelS", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["n", "2"]]]], " ", SuperscriptBox["z", "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List["-", FractionBox["n", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", SuperscriptBox["\[Pi]", FractionBox[RowBox[List["2", "+", "n"]], "2"]], " ", "z", " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List["n", "/", "2"]]], " ", RowBox[List["FresnelS", "[", RowBox[List["a", " ", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["n", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["n", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["n", "/", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["2", "+", "n"]], "2"], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "a", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]]]]]]

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

2001-10-29