Can't see what's been done in this step. (fourier cosine transform) Watch

Badges: 15
Report Thread starter 7 years ago
The question is, "Evaluate the fourier cosine transforms for the functions f and g defined on R by  f(x) = e^-^a^x cos(ax)     g(x) =e^-^a^xsin(ax)

I was looking at the solution sheet and get it until this step:

 \sqrt(\frac{2}{\pi}) \int_{0}^{\infty} e^-^a^x^(^1^-^i^) cos(sx) dx = \sqrt(\frac{2}{\pi})(\frac{a(1 - i)}{a^2(1-i)^2 + s^2}

I thought it was integration by parts to evaluate the integral except I got about halfway and it looked like the messiest thing ever and it didn't seem like I'd get the same result as written in the solutions. I kind of don't want to integrate the whole thing by parts only to realise there's a better way to do it :p: or atleast a trick that makes it simpler. Plus I feel like I'd have to integration by parts multiple times no matter which way round I put my  u and  \frac{dv}{dx}. I've even tried putting the cos into exponential form, multiplied it all out and tried integrating like that but I don't think I completely get rid off all exponentials that way. Although, I did that in a rush so I'm going to re-check that whilst I wait for a reply. Am I completely overlooking something? :confused:

EDIT: I subbed in 1 instead of 0 by accident after converting cos into exponentials. I think it should work now, probably. Oops.
Badges: 2
Report 7 years ago
Yes, it's integration by parts. Comes out in two lines, and a bit.

Let k = a(1 - i) to simplify things. Then let u = e^(-kx) and dv/dx = cos(sx)

No need to put cos into exponential form.

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