I was given the following formula for the Fourier series of a function with period $2\pi$:
$$ \begin{align*} \hat f(x) = \frac {a_0} 2 + \sum_{n=1}^\infty a_n \cos (nx) + b_n \sin (nx) \end{align*} $$
This formula is awkward because the coefficient $a_0$ is halved, whereas the coefficients $a_n$ and $b_n$ aren't, for $n > 0$. Then I realized that I could rewrite this as:
$$ \begin{align*} \hat f(x) = \sum_{n \in \mathbb Z} c_n \cos (nx) + d_n \sin (nx) \end{align*} $$
Where:
$$ \begin{align*} a_n & = c_n + c_{-n} && \mbox{for } n \ge 0 \\ b_n & = d_n - d_{-n} && \mbox{for } n \ge 0 \\ \end{align*} $$
This is IMO much easier on the eyes. Then you can add a constraint that $c_n, d_n = 0$ when $n < 0$, if you want to. Is there a good reason not to work this way?