Find the Fourier series for the function f(x) = sin(4x)

[broegger]

I have to find the Fourier series for the function f(x) = sin(4x), but no matter what I find _all_ the Fourier coefficients to be zero; i.e. $$(2\pi)^{-1}\int_{-\pi}^{\pi}sin(4x)e^{-inx}dx = 0$$ for all n.

I can't see the point in finding the Fourier series for sin(4x) anyway, since the function is simple harmonical - but shouldn't some of the coefficients be non-zero?

I'm new to the subject, so please excuse me if the answer is obvious...

 

[matt grime]

The integral of sin(4x)sin(4x) is an integral of a positive not identically zero conintuous function over some interval possibly [0,2pi], so it can' t be zero. it is a silly question, but it has shown you that there's something you're doing wrong, so it's served some purpose, surely?

 

[HallsofIvy]

Yes, ONE of the coefficients is not 0!


The point of this problem is that you should be able to recognize that it is trivial and simply write down the answer without doing any calculations!

 

[matt grime]

HallsofIvy, did you look at the latex source for that tag? seems like they're doing Fourier series the pure way not the applied way, in that they're using exp{inx} rather than sins and cosines individually, so the question isn't totally vacuous.

 

[broegger]

I don't know why my latex isn't being generated.

We're using the system {exp(inx)} - which coefficient is the non-zero one?

 

[HallsofIvy]

Thanks, Matt, I didn't look at that before.

broegger, there seems to a general problem with the latex lately.

What I meant was, assuming you were writing the Fourier series as a sum of sin(nx), cos(nx) (Matt, YOU were the one who mislead me, by talking about integrating sin(4x)sin(4x)!). In that case, sin(4x) IS the Fourier series.

Okay, since you are writing the Fourier series as a sum of e^inx, there will be two non-zero coefficients. Do you know how to write sine as exponentials?

 

[matt grime]

I missed it the first time too, and only checked cos i was wondering exactly over what interval we were working. Sorry for sending you off on the slightly wrong tangent, HallsofIvy.

 

[broegger]

Do you know how to write sine as exponentials?

Oh, of course.. sin(nx) = 1/(2i)[exp(inx)-exp(-inx)].. It turns out that the Fourier series is just sin(4x) itself.. thanks for your help!

I have a lot of exercises so I'll probably be back with more questions in this thread ;)

 

[matt grime]

In this case, expanding using exponentials the Fourier series is:

(1/2i)*(exp{4inx} - exp{-4inx})