I do understand how we obtain the Fourier Transform
$$f(x) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(\xi) e^{-i \omega \xi} d \xi \; e^{i \omega x} d \omega.$$
However, in many textbooks I see that this should be equivalent to
$$f(x) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(\xi) e^{-i 2 \pi \omega \xi} d \xi \; e^{i 2 \pi \omega x} d \omega,$$
which I do not understand. I guess this is something very simple, but I do not get there. Could someone please guide me through the math step by step?