Is there a closed formula for the Fourier transform of the function \begin{equation} f(t) = e^{2\pi i \sqrt{1-t^2}}, \end{equation} where the square root for $|t|>1$ is $i \sqrt{t^2-1}$. This function emerges in Fourier optics.

  • $\begingroup$ How do you define the square root for $|t|>1$? $\endgroup$ – Start wearing purple Sep 1 '13 at 11:39
  • $\begingroup$ I added the definition. $\endgroup$ – Guido Kanschat Sep 1 '13 at 11:54
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    $\begingroup$ This would give exponential growth at infinity. Are you sure about the sign? $\endgroup$ – Start wearing purple Sep 1 '13 at 12:09
  • $\begingroup$ Good point. It had to be the decaying function $\endgroup$ – Guido Kanschat Sep 1 '13 at 15:48

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