Finding Fourier extension and if it converges

[Weilin Meng]

Homework Statement

Let f(x) = sin(x)/x for |x| <= pi with the obvious definition at x = 0

Extend it periodically. Will the Fourier series converge at x=0?

Homework Equations

Fourier coefficients:

ao = 1/\pi \int_{-\pi}^{\pi} (sin(x)/x)

an = 1/\pi \int_{-\pi}^{\pi} (sin(x)/x) * cos(nx)

bn = 1/\pi \int_{-\pi}^{\pi} (sin(x)/x) * sin(nx)

The Attempt at a Solution

ao = 1/\pi \int_{-\pi}^{\pi} (sin(x)/x) = 2Si(pi)/pi

an = 1/\pi \int_{-\pi}^{\pi} (sin(x)/x) * cos(nx) = (-Si((n-1)pi) + Si((n+1)pi))/pi

bn = 1/\pi \int_{-\pi}^{\pi} (sin(x)/x) * sin(nx) = 0

So the Fourier series will be:
f(x)=1/\pi*Si(\pi)+\sum_{1}^{\infty }1/\pi(-Si((n-1)\pi) + Si((n+1)\pi)

at x=0 because cos(n(0)) = 1

I figured that:
\sum_{1}^{\infty }1/\pi(-Si((n-1)\pi) + Si((n+1)\pi)
Will converge to zero and we get f(x) = si(pi)/pi...

I don't know what si(pi)/pi is but I don't think that converges to 1 as sin(x)/x does at x=0...did i mess up anything?

 

[Weilin Meng]

bump, nobody yet? Am I not providing anything?

 

[Hurkyl]

I'm not too familiar with the sine integral function, but seeing how your summand consists of a difference of two evaluations (and the arguments have a fixed difference! Surely that will help simplify whatever you do), maybe there's a "difference of si's" identity you could use? Maybe the difference could be expressed usefully as a differential approximation, or even a full blown Taylor series. Or, maybe there's another way to simplify the infinite sum...

 

[Dick]

Ok, since you are bumping, do you really need to work out the Fourier series to figure out where it converges? Don't you have theorems about where it converges to save you this pain?

 

[Weilin Meng]

Ok, since you are bumping, do you really need to work out the Fourier series to figure out where it converges? Don't you have theorems about where it converges to save you this pain?

Haha, I don't know if I mentioned, but this is from a PDE course, and we are asked to compute the Fourier series and see if it converges with the original function...The course doesn't assume a huge background in math except some knowledge in ODE's, linear algebra and multi-calc. Unfortunately I have no more memory of how to do taylor or power series, but I am sure that the question does not ask for that.

Also I did not know about the Si function until I put it in an integrator..I admit that the professor usually goes crazy when coming up with his own problems...

anywho the question asks whether it converges or not...so for all I know I could be right in that it does not converge. Can anybody confirm this?