Overview
There are alternate versions of the Fourier Transform. Typically these involve where both the {% 2 \pi %} goes in the formula, and where the negative sign goes. The important thing with Fourier formulations is that whichever formulation is used, it is used consistently.
The following is a second common formulation.
{% \hat{f}(p) = \int_{-\infty}^{\infty} f(x)e^{2 \pi i p x} dx %}
{% f(x) = \int_{-\infty}^{\infty} \hat{f}(p)e^{-2 \pi i p x} dp %}