Tuesday, 24 September 2013

real analysis - Generalized Fresnel integral inti0nftysinxp,rmdx




I am stuck at this question. Find a closed form (that may actually contain the Gamma function) of the integral



0sin(xp)dx



I am interested in a Laplace approach, double integral etc. For some weird reason I cannot get it to work.



I am confident that a closed form may actually exist since for the integral:



0cosxadx=πcscπ2a2aΓ(1a)




there exists a closed form with Γ and can be actually be reduced further down till it no contains no Γ. But trying to apply the method of Laplace transform that I have seen for this one , I cannot get it to work for the above integral that I am interested in.



May I have a help?


Answer



If p>1,
I(p)=+0sin(xp)dx=1p+0x1p1sin(x)dx
but since L(sin(x))=1s2+1 and L1(x1/p1)=s1/pΓ(11p) we have:
I(p)=1pΓ(11p)+0s1/p1+s2ds=π2pΓ(11p)sec(π2p).


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