Fresnel integral S: Differential equations (formula 06.32.13.0007)

FresnelS

$Mathematica Notation$

Gamma, Beta, Erf

FresnelS[z]

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

http://functions.wolfram.com/06.32.13.0007.01

Input Form

Wronskian[h[z] FresnelS[g[z]], h[z] FresnelC[g[z]], h[z], z] == (-Pi) g[z] h[z]^3 Derivative[1][g][z]^3

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Wronskian", "[", RowBox[List[RowBox[List[RowBox[List["h", "[", "z", "]"]], RowBox[List["FresnelS", "[", RowBox[List["g", "[", "z", "]"]], "]"]]]], ",", RowBox[List[RowBox[List["h", "[", "z", "]"]], RowBox[List["FresnelC", "[", RowBox[List["g", "[", "z", "]"]], "]"]]]], ",", RowBox[List["h", "[", "z", "]"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", RowBox[List["g", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "3"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "3"]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> W </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> S </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> C </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> g </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mi> h </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <msup> <mi> g </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> h </ci> <ci> z </ci> </apply> <apply> <ci> S </ci> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> h </ci> <ci> z </ci> </apply> <apply> <ci> C </ci> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> h </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <ci> g </ci> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> h </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> g </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Wronskian", "[", RowBox[List[RowBox[List[RowBox[List["h", "[", "z_", "]"]], " ", RowBox[List["FresnelS", "[", RowBox[List["g", "[", "z_", "]"]], "]"]]]], ",", RowBox[List[RowBox[List["h", "[", "z_", "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["g", "[", "z_", "]"]], "]"]]]], ",", RowBox[List["h", "[", "z_", "]"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", RowBox[List["g", "[", "z", "]"]], " ", SuperscriptBox[RowBox[List["h", "[", "z", "]"]], "3"], " ", SuperscriptBox[RowBox[List[SuperscriptBox["g", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]], "3"]]]]]]]

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

2007-05-02