Wednesday, November 21, 2018

calculus - How to evaluate integral intinfty0ex2fracsin(ax)sin(bx)dx?

I came across the following integral:



0ex2sin(ax)sin(bx)dx



while trying to calculate the inverse Laplace transform




L1p[sinh(αp)sinh(βp)eγpp],|α|<β,γ>0



using the Bromwich integral approach. The contour I used is the following:



enter image description here



the above mentioned integral arises while doing integration over the segments L+1,L+2, and L1,L2,.



I have searched for this integral in Prudnikov et. al., Integrals and Series, v.1, but found nothing. I have also tried to evaluate the integral using residue theorem, but could not quite decide which contour to use.




Any help is greatly appreciated!



P.S. The ILT can be calculated by noticing that
F[p]=sinh(pα)sinh(pβ)eγpp=n=0(e(α+β+γ+2nβ)ppe(α+β+γ+2nβ)pp)
using
L1p[eαpp]=1πteα24t
we get
f(t)=L1p[F(p)]=n=0(e(α+β+γ+2nβ)2/4tπte(α+β+γ+2nβ)2/4tπt).
Here I am more interested in calculating the above ILT using the Bromwich integral approach.

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