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!

Summation formula for trig functions

  1. May 3, 2009 #1
    Does anyone know if there is a summation formula to find the sum of an expression with n as an argument in a trig function? I'm asking this because I'm learning about Fourier series/analysis but it seems that once we have the Fourier series we only sum for n=1,n=2,n=3... We never sum there series all the way to infinity. Is there a way to do that, simular to the way an integral is defiened as the Riemann sum of an infinate number of points?

    If anyone can point me in a good dirrection I would sincerely appreciate it.

    Thanks a lot
  2. jcsd
  3. May 3, 2009 #2

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    There are some tricks that can be used, but not every series can be summed explicitly. As an example, take

    [tex]\sum \frac1n \sin nx[/tex]

    Using complex variables, we can write this as

    [tex]\Im \sum \frac1n e^{inx}[/tex]

    Now let [itex]z = e^{ix}[/itex], so we can write

    [tex]\Im \sum \frac1n z^n[/tex]

    And now use the series for [itex]\ln (1+z)[/itex]:

    [tex]\ln (1+z) = z - \frac{z^2}2 + \frac{z^3}3 - \frac{z^4}4 + \ldots[/tex]

    to write the sum as

    [tex]\Im (- \ln (1-z)) = - \arg (1-z) = - \arctan \frac{\Im (1-z)}{\Re (1-z)}[/tex]

    Finally, put in the definition of z:

    [tex]\sum \frac1n \sin nx = \arctan \frac{\sin x}{1 - \cos x}[/tex]

    If you plot this, it reproduces the periodic ramp function represented by the original series.

    To learn how it works, why don't you try working it out for a square wave? Basically all you do is use the ln(1+z) series in creative ways.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook