Fourier cosine series of cos(x) from x=0 to Pi

[alias99]

Homework Statement

Find the Fourier cosine series for f(x) = cos x, 0 < x < Pi

EDIT: I believe we are talking about a half-range extension..

Homework Equations

Fourier cosine series:
f(x) = a₀/2 + Sum(n = 1 to infinity) (a_n * cos (nx))

where a₀ = 2/L * integral (x = 0, x = L) (f(x) dx)

and a_n = 2/L * integral (x = 0, x = L) (f(x) * cos (n*Pi*x/L) dx)

Sorry if that's hard to understand...

The Attempt at a Solution

I graphed the function and extended the period to - Pi, i.e. -Pi < x < Pi

I found a₀ = 2/Pi * integral (x = 0, x = L) (cos x dx) = 0
a_n = 2/Pi * integral (x = 0, x = L) (cos x * cos (nx) dx)
and evaluated the integral using IBP...

a_n = 2/Pi * (ncos(x)*sin (nx) - sin(x)cos(nx))*1/(n²-1)| x = 0, x = Pi
which gives me a_n = 0!

So the Fourier cosine series would be:
f(x) = 0

I would really appreciate any clarification as getting 0 for the series doesn't sound right..

Thank you in advance

 

[tiny-tim]

Hi alias99!

I found a₀ = 2/Pi * integral (x = 0, x = L) (cos x dx) = 0
a_n = 2/Pi * integral (x = 0, x = L) (cos x * cos (nx) dx)
and evaluated the integral using IBP...

a_n = 2/Pi * (ncos(x)*sin (nx) - sin(x)cos(nx))*1/(n²-1)| x = 0, x = Pi
which gives me a_n = 0!

Nooo … no need for IBP here (and it doesn't help anyway ) …

learn your https://www.physicsforums.com/library.php?do=view_item&itemid=18" …

cosx cos(nx) = … ?

 

[alias99]

Hi tiny-tim! (reminds me of the Nudie juices..hmm)
Thanks for your response ;)

You're right, I do need to learn my identities

I can now evaluate the integral much easier (Lol... funny thing is I did manage to get a result by IBP.. it looked like it would have continued indefinetely but I got the original integral on the RHS, took it over to the LHS and made it the subject)

so now..

cosx cos(nx) = 1/2 (cos (x-nx) + cos (x+nx)) which can be integrated fairly easily

I still get 0! hehe.. I guess it is 0 after all. probably has something to do with cos(x) having an area under the curve of 0 between x = -Pi and x = Pi (or even x = 0 to Pi)

I'm probably going to state that the Fourier cosine series of cos (x) with a period of Pi is cos(x) .. or is it 0.. lol

I'd appreciate just a word of confirmation, if you don't mind :)

 

[tiny-tim]

Hi alias99!

cosx cos(nx) = 1/2 (cos (x-nx) + cos (x+nx)) which can be integrated fairly easily

I still get 0! hehe.. I guess it is 0 after all. probably has something to do with cos(x) having an area under the curve of 0 between x = -Pi and x = Pi (or even x = 0 to Pi)

I'm probably going to state that the Fourier cosine series of cos (x) with a period of Pi is cos(x) .. or is it 0.. lol

I'd appreciate just a word of confirmation, if you don't mind :)

Well, it seemed a strange question …

Find the Fourier cosine series for f(x) = cos x, 0 < x < Pi

… if f is 2π-periodic, then yes obviously the only term in its Fourier cosine series is the first one, cosx.

 

[alias99]

Great! Thanks a lot tiny-tim

 

[wk1989]

I also just tried to do this and it looks like this is pretty trivial if you try to get the exponential Fourier series instead? Because cosx can be written as e^jwt+e^-jwt