Need help with solving this hard integral!

m06antwe

I have big problems solving this integration:

[tex]
\int_{-a}^{a}\sqrt{a^2-x^2}e^{i(kx-wt)}dx
[/tex]

I did an substitution with:

[tex]
x=a*sin(u)
[/tex]

Which gave me:

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

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:

[tex]
v=a*sin(u)
[/tex]

But without any success... Please help me somebody!

LCKurtz

Write it as

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

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

[tex]e^{-i\omega t}\frac{a\pi}{k}BesselJ(1,ka)[/tex]

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

m06antwe

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!