What is the difference (if any) between using:
$\cos \left( \frac{n\pi x}{L} \right)$ and $ \sin \left( \frac{n\pi x}{L} \right)$
or using:
$\cos \left( \frac{2n\pi x}{L} \right)$ and $ \sin \left( \frac{2n\pi x}{L} \right)$
In a fourier series?
My guess would be that the first approaches faster to the actual $f(x)$ (in computing for example), but in the infinity sum they are both equvalent.
Thanks.
Let $f(t)$ be a function with period $T$ then if we want to express $f(t)$ as a Fourier series
$$f(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}\left[a_n\cos(nt)+b\sin(nt) \right]$$
we can calculate the coefficients by
$$a_n = \frac{2}{T}\int_{0}^{T}f(t)\cos(nt)dt$$
$$b_n = \frac{2}{T}\int_{0}^{T}f(t)\sin(nt)dt.$$
If you take different expressions for cosine and sine you will need to adapt the formulas for the coefficients. Hence, I would always try to stick to one formula.