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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 power, exponential and hyperbolic functions > Involving power, exp and sinh





http://functions.wolfram.com/06.33.21.0094.01









  


  










Input Form





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










Standard Form





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MathML Form







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Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.