I am trying to find the Fourier Transform of the periodic function

$$ f (\theta) = \sum_{n=- \infty}^{\infty} a_n \space \exp(i n \theta)$$.

Using the formula, $$ f(k) = \int_{- \infty}^{\infty} d \theta \space \exp(-ik \theta ) f(\theta)$$

I get the following infinite sum $ 2 \pi a_n \sum_{n=- \infty}^{\infty} \space \delta (k - n)$

Where do I go from here?

  • 1
    $\begingroup$ Do you really get two $2\pi a_n$s? $\endgroup$ Dec 7, 2019 at 5:02
  • $\begingroup$ @LordSharktheUnknown sorry typo, edited $\endgroup$ Dec 7, 2019 at 5:13
  • $\begingroup$ Did you remove the correct $a_n$? $\endgroup$ Dec 7, 2019 at 5:22
  • 4
    $\begingroup$ I believe the $a_n$ should be inside the last sum. Other than that, what you've done is correct. $\endgroup$ Dec 7, 2019 at 5:28
  • $\begingroup$ How did you define the Fourier transform of a non-$L^1$ nor $L^2$ function such as $\exp(in \theta)$. Also what is the Fourier transform of $\sum_n a_n \delta(k-n)$ ? $\endgroup$
    – reuns
    Dec 7, 2019 at 7:38


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