Can you explain how they go from the first line to the second? More specifically, what are the steps for making the negative exponential terms positive?

i.e. how does the third term: $-e^{j(4\pi t + \pi/3)}$ become $e^{-j(2\pi /3)} e^{j4\pi t}$ ?

enter image description here

This step shown above is part of a bigger question on finding Fourier Series.

  • $\begingroup$ would this be a correct explanation as well: since there is a neg in front of cos we invert the wave about the x axis. Then to get back to the original cos wave we have to shift right by pi. Thus we do cos (4*pi*t + pi/3 - pi)?? I just thought about this right now, please confirm if the logic is correct. $\endgroup$ – rrazd Aug 31 '12 at 1:32

Remember that $e^{j \pi} = e^{-j\pi} = -1$. So $e^{j\pi/3} = - e^{-j\pi} e^{j\pi/3} = -e^{-j2\pi/3}$.


$$\frac{1}{2}e^{j(4\pi t+\pi/3)}e^{-j2\pi+j2\pi}=\frac{1}{2}e^{j4\pi t}e^{-j2\pi/3}e^{j2\pi}=\frac{1}{2}e^{j4\pi t}e^{-j2\pi/3}$$

as $e^{j2\pi}=1$


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