## anonymous 3 years ago Evaluate the following integral: cos((mpix)/L)cos((npix)/L) for (-L, L) where m≠n, and m,n are integers

1. anonymous

$\cos \frac{ m Pix }{ L }\cos \frac{ n Pix }{ L }$

2. TuringTest

Fourier?

3. anonymous

what is fourier?

4. TuringTest

It's what you are leading up to studying, though apparently your teacher has not told you yet use$\cos\alpha\cos\beta=\frac12[\cos(\alpha-\beta)+\cos(\alpha+\beta)]$to get this integral manageable.

5. anonymous

what is alpha and what is beta? m and n? or L and -L?

6. TuringTest

$$\alpha$$ and $$\beta$$ are just the arguments of the cosines:$\alpha=\frac{m\pi x}L$$\beta=\frac{n\pi x}L$

7. anonymous

so do we then take the integral of (1/2)[cos(alpha-beta)+cos(alpha+beta)]?

8. TuringTest

yes

9. anonymous

but how would i integral the inside? (alpha-beta)?

10. anonymous

sin(alpa)cos(beta)?

11. TuringTest

alpha and beta just represent any argument in the trig identity I am showing you substitute$\alpha=\frac{m\pi x}L$$\beta=\frac{n\pi x}L$

12. anonymous

so wouldnt the integral be $\sin( \alpha)$$\cos (\beta)$

13. anonymous

multiplied together that is?

14. TuringTest

no, I have no idea where you are getting that :p $\cos\alpha\cos\beta=\frac12[\cos(\alpha-\beta)+\cos(\alpha+\beta)]$sub for $$\alpha$$ and $$\beta$$ as I have stated above twice before

15. TuringTest

you get$\cos\left(\frac{m\pi x}L\right)\cos\left(\frac{n\pi x}L\right)=\frac12[\cos\left(\frac{\pi x}L(m-n)\right)+\cos\left(\frac{\pi x}L(m+n)\right)$which you can integrate

16. anonymous

ok, just a second, let me give that a try

17. TuringTest

also, because this integral is even we can write$\int_{-L}^L\cos\left(\frac{m\pi x}L\right)\cos\left(\frac{n\pi x}L\right)dx=2\int_0^L\cos\left(\frac{m\pi x}L\right)\cos\left(\frac{n\pi x}L\right)dx$

18. anonymous

(1/2pi)(L((sin(pix(m-n)/L)/(m-n))+sin(pix(m+n)/L)/(m+n))

19. anonymous

is that right?

20. TuringTest

It's hard for me to see if all those parentheses are in the right places, but it looks right I'm afraid I have to go. Here's a link to the problem you are solving: http://tutorial.math.lamar.edu/Classes/DE/PeriodicOrthogonal.aspx#BVPFourier_Orthog_Ex1 Look at example 1 (the last part of it) for reference Good luck!

21. anonymous