Sometimes when deriving the formulas for the coefficients of Fourier series mathematicians start with this definition:
$$f(t):=a_0+\sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi t}{L}+b_n\sin\frac{n\pi t}{L}\right]$$
But other times they start with:
$$f(t):=a_0+\sum_{n=1}^{\infty}\left[a_n\cos nt+b_n\sin nt\right]$$
The second one seems more intuitive but what's the intuition behind the first one? Are they equivalent?
The only thing that differs is the domain (or the period). In the former case, $f:[-L,L]\to\mathbb{C}$ while in the latter case, $f:[-\pi,\pi]\to\mathbb{C}$.
They are practically equivalent. One can easily transform one case into the other by scaling $f$ in the $x$-direction.