Question

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

Answer 1

\[\cos \frac{ m Pix }{ L }\cos \frac{ n Pix }{ L }\]

Answer 2

Fourier?

Answer 3

what is fourier?

Answer 4

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.

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

yes

Answer 9

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

Answer 10

sin(alpa)cos(beta)?

Answer 11

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

Answer 12

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

Answer 13

multiplied together that is?

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

is that right?

Answer 20

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!

Answer 21

ok thank you for your patient and your help TurningTest

Answer 22

patience*