5
$\begingroup$

I studied fourier series as an undergrad and grad. student in EE but did not fully grasp the concepts. Now that I am involved in medical imaging (MRI) understanding the basics of fourier series and transforms is very important and I am frustrated at my level of understanding. For example, a problem in a MRI physics book asks:

Prove the following fourier series identity: $$\sum_{n=-\infty}^\infty \exp(2i\pi n a) = \sum_{m=-\infty}^\infty \delta(a-m)$$. The author gives a hint: "consider integrations over small intervals that either include or exclude the region where the argument of one of the delta functions vanishes".

My approach was to write out the left side of the equation for n = -5 to 5. I end up with the value 11. something's wrong for sure. Any help is greatly appreciated!

Thank you -Dave

  • 1
    Strictly speaking, that is not a Fourier series. Did you ever read about distribution theory?2011-01-06
  • 0
    Can you please give us the book title and the author?2011-01-06
  • 0
    How did you get an $11$ when both sides of the equation depend on the variable $a$?2011-01-06
  • 0
    The hint suggests you integrate both sides from $p-\epsilon$ to $p+\epsilon$ for small $\epsilon$ and 1)a particular integer $p$, then for a non-integer $p$ (so no integers are in the interval of integration). I'll do the RHS for non-integer $p$-it is zero. Do you see why? Do you see why this might help?2011-01-06

2 Answers 2