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FresnelC




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelC[z] > Integration > Indefinite integration > Involving direct function and Gamma-, Beta-, Erf-type functions > Involving erf-type functions and a power function > Involving erfc and power





http://functions.wolfram.com/06.33.21.0124.01









  


  








Input Form





Integrate[z^(\[Alpha] - 1) Erfc[b z] FresnelC[a z], z] == (z^\[Alpha]/\[Alpha]) FresnelC[a z] (Erfc[b z] + (b z Gamma[(1 + \[Alpha])/2, 0, b^2 z^2])/((b^2 z^2)^((\[Alpha] + 1)/2) Sqrt[Pi])) + (((2^((1/2) (-3 + \[Alpha])) a z^(1 + \[Alpha]))/ (Pi^((\[Alpha] + 1)/2) \[Alpha])) ((I a^2 z^2)^((1 + \[Alpha])/2) Gamma[(1 + \[Alpha])/2, (-(1/2)) I a^2 Pi z^2] + ((-I) a^2 z^2)^((1 + \[Alpha])/2) Gamma[(1 + \[Alpha])/2, (1/2) I a^2 Pi z^2]))/(a^4 z^4)^((\[Alpha] + 1)/2) + 2 a Sum[(((-1)^k 2^(-1 + k + \[Alpha]/2) b^(1 + 2 k) Pi^(-k - (\[Alpha] + 3)/2) z^(2 + 2 k + \[Alpha]))/ ((1 + 2 k) (1 + 2 k + \[Alpha]) k!)) ((I a^2 z^2)^(-1 + k + \[Alpha]/2) (a^4 z^4)^(-k - \[Alpha]/2) Gamma[\[Alpha]/2 + k + 1, (-(1/2)) I a^2 Pi z^2] + ((-I) a^2 z^2)^(-1 + k + \[Alpha]/2) (a^4 z^4)^(-k - \[Alpha]/2) Gamma[\[Alpha]/2 + k + 1, (1/2) I a^2 Pi z^2]), {k, 0, Infinity}]








Standard Form





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MathML Form







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</mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Erfc </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <apply> <plus /> <ci> &#945; 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</ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </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> <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 /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> &#945; 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Rule Form





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





2001-10-29