Fourier Integral Proof question

[iocal]

Hi all,

I am looking for some help in proving the following:

\(\displaystyle \displaystyle \int_0^{\infty} {\left [A(\lambda)\cos\lambda x +B(\lambda)\sin\lambda x \right ]}\ \mathrm d\lambda\)

with

\(\displaystyle \displaystyle A(\lambda) = \frac 1\pi \int_{-\infty}^{\infty} F(x)\cos \lambda x\ \mathrm dx\)

\(\displaystyle \displaystyle B(\lambda) = \frac 1\pi \int_{-\infty}^{\infty} F(x)\sin \lambda x\ \mathrm dx\)

can be rewritten equivalently as:

\(\displaystyle \displaystyle \frac 1{2\pi}\int_{\lambda=-\infty}^{\infty} \int_{u=-\infty}^{\infty} F(u)\ \cos\lambda (x-u)\ \mathrm du\ \mathrm d\lambda\)

I know that most likely it can be shown by plugging the expressions for A and B into the Fourier Integral but it gets very messy at some point and I get lost. And of course I am familiar with the trigonometric identity of the cosine.
If you reply please talk to me like I am a beginner in Fourier Analysis because I really am!
Thank you.

 

[HallsofIvy]

Start by using the fact that \(\displaystyle cos(x- \mu)= cos(\mu)cos(x)+ sin(\mu)sin(x)\).

And then do the integral with respect to \(\displaystyle \mu\).