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FresnelC






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelC[z] > Integration > Indefinite integration > Involving one direct function and elementary functions > Involving hyperbolic functions and a power function > Involving sinh and power





http://functions.wolfram.com/06.33.21.0074.01









  


  










Input Form





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










Standard Form





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MathML Form







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</mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mi> &#945; </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; 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</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> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> &#945; </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; 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Rule Form





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





2001-10-29





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