Addition of exponentials, and relationship between variables.

[Animastryfe]

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.

[I like Serena]

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:

[Animastryfe]

Thank you. I should think more geometrically.

[HallsofIvy]

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.