# Fourier Transform : complex or real?

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,

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

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.