Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If I have an equation such as $x(t) = \displaystyle \sum_{n=1}^N \left( a_n \cos(\omega_nt) + b_n \sin(\omega_n t) \right)$, how do I convert it to a sum of complex exponentials? In other words what do I do with the coefficients in front of the sine and cosine to turn them into the coefficient of each complex exponential.

share|cite|improve this question
up vote 3 down vote accepted

Write $\displaystyle \cos(w_n t) = \frac{e^{i w_n t} + e^{-i w_n t}}{2}$ and $\displaystyle \sin(w_n t) = \frac{e^{i w_n t} - e^{-i w_n t}}{2i}$ and rearrange to convert it to a sum of complex exponentials.

share|cite|improve this answer
so if I had 2cos(wt) + 4jsin(wt), what would be the resulting complex exponential? – mike Aug 14 '11 at 6:48
It will be $$3e^{j\omega t} - e^{-j\omega t}$$ – user17762 Aug 14 '11 at 7:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.