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Addition of exponentials, and relationship between variables.

  1. Aug 2, 2011 #1
    1. The problem statement, all variables and given/known data
    This is, strictly speaking, not a homework question. I have already solved this, but I think that there is a much better method to solve it.

    In the equation below, what relationship must w and q satisfy? If the question is not clear, please read the bottom of the post.

    2. Relevant equations
    Exp[-i*q*t]+Exp[-i*w*t]=-2


    3. The attempt at a solution
    I turned everything into cosines and sines, and used the trigonometric sum to product formulas.

    In case the question isn't clear, the answer is w/q= (2m-1)/(2n-1), where m and n are positive integers and not equal to each other.
     
  2. jcsd
  3. Aug 2, 2011 #2

    I like Serena

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    Homework Helper

    Hi Animastryfe! :smile:

    Assuming q and w are real numbers, the two exponentials each correspond to a vector with length 1 and angle -qt respectively -wt.
    This is the polar coordinate representation of a complex number.

    To get them to have sum -2, both the exponentials must come out as -1.
    This means that -qt = pi mod 2pi and that -wt = pi mod 2pi.
    Divide them on each other and you get the result you have.

    However, if q and w can be imaginary as well, you get a lot more solutions! :wink:
     
  4. Aug 2, 2011 #3
    Thank you. I should think more geometrically.
     
  5. Aug 3, 2011 #4

    HallsofIvy

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    Please note that, in general, for two complex numbers to add to -2, they do not have be each be -1. Here, however, each has magnitude 1 so in terms of a vector addition, we have a "triangle" with two sides of length 1 and the third of length 2. That, of course, is impossible except in the very special case that the "triangle" is really a straight line.
     
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