# Need help with solving this hard integral!

## Main Question or Discussion Point

I have big problems solving this integration:

$$\int_{-a}^{a}\sqrt{a^2-x^2}e^{i(kx-wt)}dx$$

I did an substitution with:

$$x=a*sin(u)$$

Which gave me:

$$\int_{-{pi}/2}^{{pi}/2}a^2cos^2(u)e^{i(kasin(u)-wt)}du$$

But i don't know if that did it any better, cause i can't figure out how to go on from there. I've been told to try the substitution:

$$v=a*sin(u)$$

## Answers and Replies

LCKurtz
Homework Helper
Gold Member
I have big problems solving this integration:

$$\int_{-a}^{a}\sqrt{a^2-x^2}e^{i(kx-wt)}dx$$
Write it as

$$e^{-i\omega t}\int_{-a}^a \sqrt{a^2-x^2}\cos(kx) + i\sqrt{a^2-x^2}\sin(kx))\, dx$$

The second term is an odd function of x which contributes nothing to the answer. The first term isn't going to give you an elementary answer. Maple gives

$$e^{-i\omega t}\frac{a\pi}{k}BesselJ(1,ka)$$

where BesselJ is the Bessel function of the 1st kind of index 1 with argument ka.

Oh, I would not have figured out that myself! I've never seen the Bessel function before, but it seems to give me the same answer when I'm solving it numerically in Matlab so therfore I'm happy!

Thanks alot, that really helped me!