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http://functions.wolfram.com/06.32.07.0002.01
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FresnelS[z] == (1/Sqrt[2 Pi]) Integrate[Sin[t]/Sqrt[t],
{t, 0, (Pi z^2)/2}] /; Abs[Arg[z]] < Pi/4
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FresnelS", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z", "2"]]], "2"]], RowBox[List[FractionBox[RowBox[List["Sin", "[", "t", "]"]], SqrtBox["t"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "4"]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <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> <mn> 1 </mn> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 2 </mn> </mfrac> </msubsup> <mrow> <mfrac> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <msqrt> <mi> t </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mfrac> <mi> π </mi> <mn> 4 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> FresnelS </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <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> </uplimit> <apply> <times /> <apply> <sin /> <ci> t </ci> </apply> <apply> <power /> <apply> <power /> <ci> t </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <arg /> <ci> z </ci> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FresnelS", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z", "2"]]], "2"]], RowBox[List[FractionBox[RowBox[List["Sin", "[", "t", "]"]], SqrtBox["t"]], RowBox[List["\[DifferentialD]", "t"]]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "4"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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