Register to reply

Need help with solving this hard integral!

by m06antwe
Tags: integral, solving
Share this thread:
m06antwe
#1
Feb6-11, 12:24 PM
P: 2
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!
Phys.Org News Partner Science news on Phys.org
Flapping baby birds give clues to origin of flight
Prions can trigger 'stuck' wine fermentations, researchers find
Socially-assistive robots help kids with autism learn by providing personalized prompts
LCKurtz
#2
Feb6-11, 01:47 PM
HW Helper
Thanks
PF Gold
LCKurtz's Avatar
P: 7,659
Quote Quote by m06antwe View Post
I have big problems solving this integration:

[tex]
\int_{-a}^{a}\sqrt{a^2-x^2}e^{i(kx-wt)}dx
[/tex]
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
#3
Feb6-11, 02:14 PM
P: 2
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!


Register to reply

Related Discussions
Very hard x-solving problem... Precalculus Mathematics Homework 11
Solving 2 equation with 2 variables (hard) Calculus & Beyond Homework 10
Help in hard model of RABR - solitons- solving PDE Differential Equations 0
Having a hard time solving last part Introductory Physics Homework 1
Solving a hard equation General Math 1