Fourier transform question (optics)

[wolf party]

Homework Statement

a light source a(x) is defined by

a(x) = Acos(pi*x/a)[theta(x+(a/2)) -theta(x-(a/2))]

calculate the diffraction pattern I(X)

Homework Equations

I(X)=2pi|a^~((2pi/(LAMBDA*d))*X)|²

this is the equation for a diffraction pattern on a screen at distance d from a 1D souce
of light with wavelength LAMBDA, where a^~(k) is the Fourier transform of a(x)

The Attempt at a Solution

using the Fourier transform

a^~(k) = A/sqrt(2pi)(integrate)[exp^(-ikx) * cos((pi*x)/a)]

(integrating over the range -a/2 to a/2)

i then use euler's method to change cosine term into exponential terms, then simplify to get

= A/2*sqrt(2pi)(integrate)[exp^((i*pi*x)/a)-(ikx)) +exp^(-(i*pi*x)/a)-(ikx))]

(integrating over the range -a/2)

the answer i get once i have simplified won't simplify down to trig terms and so makes no sense when i put the value of a^~(k) into the equation for I(X). how can this integral be performed elegantly to give a decent looking answer? any help would be great

 

[wolf party]

anyone?