How Do You Prove the Integral of Cosine Functions Using Exponential Forms?

[jlmac2001]

The question asks to prove the following by writing each sine and cosine function as a sum of exponentials of arguments inwt or imwt:

integral from -pi/w to pi/w (cos (nwt)cos(mwt) dt) = {0 for n not equal to m, pi/w for n=m not equal to 0 , 2pi/w for n=m=0

Would I write the cosine funtion in the exponential form e^x+e^-x/2 and then solve by using different valuse for n and w for the one the is equal to 0 when n is not equal to m? I'm confused.

 

[arildno]

Split the job in two cases
1. You are to show that the integral is zero whenever n is not m.
2. You are to show that the integral is the value indicated whenever n=m
If you are unfamiliar with the exponential form, you might use:
cos(nx)cos(mx)=1/2(cos((n-m)x)+cos((n+m)x))

 

[jlmac2001]

example

I still don't get it. Can someone show me an example? Do i have to chose two different numbers for n and m?

 

[matt grime]

You must do it for general n and m. Specifying them for a couple of values does not prove the result for all n and m. Just leave the n and m in there as letters (constants) and do the integration.