Infinite sum over exponentials

[ThomasVogt]

Hi,

I am looking at waves that go around a ring (monochromatic solution) and got stuck with the following expression:

$\sum_{n=-\infty}^{+\infty} e^{-2\pi i n k}$

where k is an angular wavenumber that can take any real value. Anyone got an idea how to approach this?

Thanks.

 

[uart]

Hi Thomas. That's actually a "Dirac comb", it occurs a lot in sampling theory.

See: http://en.wikipedia.org/wiki/Dirac_comb

 

[Mute]

One way to see the result uart posted is as follows:

Split the sum up into two sums, one from -infinity to -1 and one from 0 to infinity, and then change variables in the first sum from n to -n. Both series are just geometric series, so you can perform the two sums using the geometric series formula and add the results together. The result you find might be unexpected to you. However, you're not done there - there is are particular values of k for which you can't use the geometric series formula. What happens to the sum at those values of k?

 

[ThomasVogt]

Uart - of course, how could I not see it! Sometimes you need someone else to point out the obvious...Thanks.

Mute, yes in my desperation I did try to do the geometric sum by taking n into the exponent but I thought you can only do this when the term under the exponent has an abs value <1? I did look at the case where the angular wavenumber \nu is an integer and of course the sum diverges for these values (can you believe I still didn't see the Dirac??). To add further to my ignorance I did actually see the pattern experimentally in the amplitude sprectrum...aaargh

Thanks again to both.

 

[CellCoree]

Hello,

The expression you have provided is known as an infinite sum over exponentials, also known as a geometric series. In this case, the exponentials have a common ratio of e^{-2\pi i k}, which can be written as a complex number with magnitude 1 and angle -2\pi k. This means that each term in the series is a complex number with magnitude 1 and angle -2\pi k times the index n.

To solve this sum, we can use the formula for the sum of a geometric series, which is given by:

S = a / (1-r)

where a is the first term and r is the common ratio. In this case, a = 1 and r = e^{-2\pi i k}, so the sum becomes:

S = 1 / (1 - e^{-2\pi i k})

Since the magnitude of e^{-2\pi i k} is 1, the denominator becomes 2, and the final result is:

S = 1/2

Therefore, the infinite sum over exponentials in your expression is equal to 1/2. I hope this helps you with your analysis of waves around a ring. Let me know if you have any further questions.

Best,