Find a Fourier Series representation

[math_trouble]

I'm having problem finding the representation for the Fourier series with

function f of period P = 2*pi such that f (x) = cosαx, −pi ≤ x ≤ pi , and α ≠ 0,±1,±2,±3,K is a
constant.

Any help is appreciated...

 

[tiny-tim]

Hi math_trouble!

You probably need one of the standard trigonometric identities for cosAcosB.

Anyway, show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[g_edgar]

The Fourier coefficients are certain integrals. Write them down. If you can't do them, let us know.

 

[math_trouble]

i evaluated a₀,a_n and b_n term
i get a₀=0

a_n= 0 when a not equals to n & 1 when a equals to n

b_n= 0

i know the general Fourier series representation is in :
a₀/2 + SUM(a_ncos(nx) + b_nsin(nx))

but then I am stuck on how to apply the general term to this case

 

[uart]

Hi Math_Trouble. Just to save confusion, can we call the parameter \alpha instead of "a", so as not to confuse it with the a_n Fourier coefficients.

Your a_n and b_n are correct for integer values of the parameter \alpha, but I thought that you wanted an expression that is valid for non-integer alpha. I'm a little confused here because you say that you've "evaluated the terms" which implies that you have an expression to evaluate. If so then where is your expression and is it valid for non-integer \alpha?

The fact is that for integer values of alpha you don't even need to do the Fourier integrals to determine the series coefficients. You can do it "by inspection" since the waveform is already a perfect cosine wave. It's not that the Fourier integrals don't work for integer \alpha, they do, it's just that the problem is not really interesting for that case (which is why I presume that they explicitly called for a non integer alpha in the question).

If you just do the Fourier integrals then you should get an expression that is valid for real (integer and non integer) values of the parameter (though you may need to take limits to evaluate the integer cases). Show us your working so far and we can help you.

 

[math_trouble]

Hi uart..thx for reminding tat \alpha is not an integer

but now I am having trouble again to write down the series coefficient for \alpha \neq integer because it seems to be too many values and not like a general expression could express them all