A Constant Fourier Series

A constant function has entirely zero Fourier coefficients except for $a_0$ , since this is the term that captures the average value of the function without any spatial variation. Thus $a_3$ vanishes.

Expanding on Matt's solution, the function is $f(x) = 1$ for $x \in [0,1]$ and period $P = 1$ . The third Fourier coefficient $a_{3}$ can be computed according to:

$a_{n} = \frac{2}{P} \cdot \int_{0}^{P} f(x) \cdot \cos(\frac{2\pi nx}{P}) dx$ ;

or $a_{3} = 2\int_{0}^{1} \cos(6\pi x) dx$ ;

or $a_{3} = \frac{1}{3\pi} \sin(6\pi x)|_{0}^{1};$

or $a_{3} = \frac{1}{3\pi} [\sin(6\pi) - \sin(0)] = \boxed{0}.$