I'm currently studying Fourier series. I'm having trouble finding the right expressions for coefficients $a_0, a_n$ and $b_n$. I have 3 sources and 3 different expressions:
My lecture notes states:
$$a_n = \frac{1}{T} \int^T_0 f(t)\cos(\frac{2\pi n}{T}t)dt$$
$$b_n = \frac{1}{T} \int^T_0 f(t)\sin(\frac{2\pi n}{T}t)dt$$
This website says:
$$a_0 = \frac{1}{T} \int^T_0 f(t)dt$$
$$a_n = \frac{2}{T} \int^T_0 f(t)\cos(\frac{2\pi n}{T}t)dt$$
$$b_n = \frac{2}{T} \int^T_0 f(t)\sin(\frac{2\pi n}{T}t)dt$$
and this other shows both expressions for $a_n$ and $b_n$ as correct:
with real part:
$$a_0 = \frac{1}{T} \int^T_0 f(t)dt$$
$$a_n = \frac{2}{T} \int^T_0 f(t)\cos(\frac{2\pi n}{T}t)dt$$
$$b_n = \frac{2}{T} \int^T_0 f(t)\sin(\frac{2\pi n}{T}t)dt$$
without real part:
$$a_0 = \frac{1}{T} \int^T_0 f(t)dt$$
$$a_n = \frac{1}{T} \int^T_0 f(t)\cos(\frac{2\pi n}{T}t)dt$$
$$b_n = \frac{1}{T} \int^T_0 f(t)\sin(\frac{2\pi n}{T}t)dt$$
It is clear that this eqs are not equivalent.
so...
- What are the correct expressions?
- What it refers when says "with real part" and "without real part"?