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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.

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  • $\begingroup$ Suppose I want to find the Fourier Series for $sin^2x$. Then what is the exact formula do I use to find $a_0$, $a_n$ and $b_n$? Also, I want know the difference between $sin(nw_0x)$ and $sin(nx)$. Thanks! $\endgroup$ – user119186 Jan 19 '14 at 6:44
  • $\begingroup$ It's really important that I understand this, please. Any help would be appreciated, thank you. $\endgroup$ – user119186 Jan 19 '14 at 7:09
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The most general formula is $f(t) = \frac{a_0}{2}+\sum_{n=1}^{\infty}{a_n\cos(n\omega t)+b_n\sin(n\omega t)}$ then you can derive $a_n = \frac{2}{T}\int_c^{c+T}f(t)\cos(nwt)dt$ and $b_n = \frac{2}{T}\int_c^{c+T}f(t)\sin(nwt)dt$. Now $T$ is the period of the function. $\omega=\frac{2\pi}{T}$ and c is an arbitrary value that you choose for your integartion e.g. your function is $2\pi$ periodic between $[-\pi,\pi]$ then you choose $c = -\frac{T}{2}=-\frac{\pi}{2}$

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