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How to prove Euler's Formula $e^{i\theta} = (cos\theta + isin\theta)$?

I know this is kind of basic and I am familiar with this equality for a long time. But, how do I prove it?

And another question: I have read that $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ is a harmonic function of the form $e^{-in\phi_i}$, and therefore it's eigenvalues are one of the Fourier components of $(|\phi_i-\phi_j|)$.

My questions are:

  1. How come $ce^{-\frac{\phi_i-\phi_j}{\rho}}$ has the form of $e^{-in\phi_i}$ if it is a real function?
  2. How do I cacaulate the eigenvalues of such a function, or, if it is a difficult calculation, is there a known theorem(s) I can base this result on?

I am not an expert in harmonic analysis, but I would like to know more about this subject (without getting into too much details). I need it to my research in neurosciences.

Here is a link to the article I am reffering to: Population coding

Thank You!

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user135172
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    See http://math.stackexchange.com/questions/3510/how-to-prove-eulers-formula-eit-cos-t-i-sin-t and https://proofwiki.org/wiki/Euler's_Formula – lab bhattacharjee Jan 11 '16 at 11:35
  • your nearly didn't post anything of your book, so it is impossible to really understand it. and what you need in my opinion is to understand the matrix point of view on the discrete fourier transform (simply an unitary matrix which sends a vector to the basis of the complex exponentials, and thus allows filtering by convolution) – reuns Jan 11 '16 at 11:51
  • I see.. where can I find this? i mean the matrix point of view on the discrete fourier transform.. – user135172 Jan 11 '16 at 12:08

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