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Complex exponential expressions.

  1. Aug 31, 2012 #1
    1. The problem statement, all variables and given/known data

    I just need some kind of explanation in layman's terms of what exactly is going on here. It seems as though I am missing some key element from trig. I am in a Signals class and the book lacks an explanation of the reduction used and ultimately why.

    2. Relevant equations

    I have attached a .pdf of the files.
    View attachment Document1.pdf
    3. The attempt at a solution

    Ok, so in "b": I do not understand the simplification or reduction to the side, where 4pi is added. Also I think the same method occurs in the third answer of "d".

    Also, is the reason there are 3 answers for "d" because of the cube root? If someone could just provide a short explanation as to why, that would be helpful.


    Thank you so much in advance!
     
  2. jcsd
  3. Sep 1, 2012 #2

    Mark44

    Staff: Mentor

    What they're doing in b is adding multiples of 2##\pi## so as to get a positive angle.

    From Euler's formula, eix = cos(x) + i*sin(x). This represents a complex number whose magnitude is 1, and that makes an angle of x radians measured counterclockwise from the positive Real axis. Note that mathematicians use i for the imaginary unit, and engineers use j.

    Adding 2##\pi## or multiples of 2##\pi## to the exponent to get ei(x + 2##\pi##) doesn't change anything except the angle. This complex number still has a magnitude of 1 - the only difference is that the angle is now x + 2##\pi##, which gets you to exactly the same point on the unit circle.


    In the same way that a number has two square roots, it will have three cube roots, four fourth roots, and so on. A real number, such as 8 has one real cube root (2) and two complex cube roots.
     
  4. Sep 1, 2012 #3
    So, it is just commonplace to work within the unit circle? I mean for my calculator both arguments produce the same answer, I just appropriate the negative when using Euler's formula.

    Thanks again.

    Oh and by the way, I am on my way to be an EE so its "j" for me -here on out!
     
  5. Sep 1, 2012 #4

    Bacle2

    User Avatar
    Science Advisor

    The complex exponential is periodic with (argument) period equal to 2Pi. So if you

    go around n times, then ez= ez+2n∏ for n=1,2,...... and

    ez+2n∏ is equivalent to going around a circle n times.
     
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