EDIT: I need help today, please. It's very important for my homework. I need to understand this concept. Thank you!
I have a doubt regarding the Fourier series formula.
In one of my notes, it is given that, for a function $f(x)$ defined on an interval $-\pi$ to $\pi$, the fourier series can be found as follows:
The series is:
$$ f(x) = a_0 + \sum_{n=1}^\infty(a_n\cos nx) + \sum_{n=1}^\infty(b_n \sin nx)$$
And that:
$$\begin{align}a_0 &= \dfrac 1{2\pi}\int_{-T_0}^{T_0}f(x)dx \\ a_n &= \dfrac 1\pi\int_{-T_0}^{T_0}f(x)\cos(nx)dx \\ b_n &= \dfrac 1\pi\int_{-T_0}^{T_0}f(x)\sin(nx)dx \end{align}$$
However, when I searched online, I found different formulae. In some of them, the integral limits went from $0$ to $\pi$, in others from $-T_0$ to $T_O$. Also, the fraction before the integral differed - in some cases, the integral was multiplied not with $\dfrac 1{2\pi}$, but $\dfrac 1T$, etc.
I need the formula to solve the fourier series for basic functions such as $\sin^2(x)$, etc.
My biggest confusion lies with regard to the 'n' term used in the formulae. In some formula, I have seen it written as, for example:
$$ f(x) = a_0 + \sum_{n=1}^\infty(a_n\cos(nw_0x)) + \sum_{n=1}^\infty(b_n\sin(nw_0x))$$
What is the difference between $nx$ and $nw_0x$, and which one do I use to solve for the fourier series? Could you please list the three formula exactly as I need? I wish to use only the trigonometric Fourier series, please.
Thanks.