1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Complex Summation

  1. Jul 13, 2009 #1
    This is not really a homework problem, but I'm studying a text, and I came across this:

    http://img198.imageshack.us/img198/4586/sumh.jpg [Broken]

    I know how to get that fraction with the exponents in it (using a summation formula). But for the life of me, I can't figure out how to manipulate that fraction to give the final result.

    For example, if I put k=0 into that fraction, I get 0/0, not 5. I tried a bunch of manipulation of the fraction to get sines and cosines, and make the denominator real, but I still can't get a closed form solution that gives the final result.

    What am I missing?


    UPDATE: Forgot to mention, this is a discrete time function. k is always an integer.
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jul 13, 2009 #2
    I totally forgot about that factoring trick:

    1-e^{-j x} = e^{-j x/2}(e^{j x/2}-e^{-j x/2}) = e^{-j x/2}jsin(x/2)

    That's all I needed!


    Wait, I was wrong. :frown: :frown:

    Even with that factoring trick, I get:

    e^{-j(\pi k-\pi k/5)}*sin(\pi k)/sin(\pi k/5)

    But this is still 0/0 for k=0. How do I get the real result?? I'm so frustrated with this! :confused:
    Last edited: Jul 13, 2009
  4. Jul 13, 2009 #3
    No matter what k is, the top is 0. If k is not 0,+/-5,+/-10,..., then the bottom is not 0, so that is how they get 0 for otherwise. For k=0,+/-5,+/-10,..., you need to find the limit as k approaches those values, because 0/0 has no meaning. Use l'Hospital's rule to evaluate your function at those points.
  5. Jul 13, 2009 #4
    Thanks, that makes sense.

    Also, instead of going to L'Hospital's rule, I could just go back to the summation for k=0,+-5, etc and show that it is a summation of ones.... while the fraction would prove the "0 otherwise" for the other k values. This would work too.

    Thanks a lot!!
  6. Jul 13, 2009 #5
    No problem. Yea, you're right about the summation giving you the five.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook