Overview
The Fourier Transform of the Top Hat function (see below) is calculated as
{% \hat{f}(p) = \int_{-\infty}^{\infty} \Pi_a(x) e^{2 \pi i p x} dx %}
{% = \int_{-a/2}^{a/2} e^{2 \pi i p x} dx %}
{% \frac{a sin(\pi p a)}{\pi p a} = a sinc(\pi p a) %}
where the sinc function is given below.
Top Hat
{% \Pi_a(x) = 1 %}
for {% \frac{-a}{2} < x < \frac{a}{2} %} and {% 0 %} otherwise.
The Fourier Transform is (using formulation)
Sinc
The sinc function is defined as
{% sinc(x) = \frac{sin x}{x} %}