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Consider a function $f\in L^1(R)$, and its Fourier transform :

$$\mathcal{F}[f](k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{iks}f(s)ds$$

I would like to know if $k$ is a complex number or if it is real. Thank you,

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  • $\begingroup$ It is real. That way you can be sure the integral always exists. The definition can also be generalized to $\mathbb{R^n}$. $\endgroup$
    – Mark
    May 15, 2020 at 9:56

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The argument $k$ is always a real number in the Fourier transform. It represents the frequency at which we want to evaluate the FT of the original signal.

If you have a problem involving non-real $k$ you're probably looking at the Laplace transform.

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