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FresnelS






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelS[z] > Integration > Indefinite integration > Involving direct function and other elementary functions > Involving trigonometric functions > Involving sin





http://functions.wolfram.com/06.32.21.0009.01









  


  










Input Form





Integrate[(Sin[b Sqrt[z]]/Sqrt[z]) FresnelS[a Sqrt[z]], z] == (1/b) (((1/4 + I/4) (E^((I b^2)/(a^2 Pi)) Erf[((1/2 + I/2) (-b + a^2 Pi Sqrt[z]))/(a Sqrt[Pi])] + E^((I b^2)/(a^2 Pi)) Erf[((1/2 + I/2) (b + a^2 Pi Sqrt[z]))/ (a Sqrt[Pi])] + Erfi[((1/2 + I/2) (b - a^2 Pi Sqrt[z]))/ (a Sqrt[Pi])] - Erfi[((1/2 + I/2) (b + a^2 Pi Sqrt[z]))/ (a Sqrt[Pi])] - (4 - 4 I) E^((I b^2)/(2 a^2 Pi)) Cos[b Sqrt[z]] FresnelS[a Sqrt[z]]))/E^((I b^2)/(2 a^2 Pi)))










Standard Form





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MathML Form







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&#8290; </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, FresnelS] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </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> <times /> <apply> <sin /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> FresnelS </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> 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</cn> </apply> </apply> </apply> <apply> <ci> Erf </ci> <apply> <times /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Erf </ci> <apply> <times /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Erfi </ci> <apply> <times /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <pi /> <cn 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Rule Form





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





2001-10-29