Question

Find the fourier series for the function f(x)= x on the interval 0≤x≤2π ?

Answer 1

$$f(x) = x, \quad x \in [0, 2\pi]$$

The fourier series is:

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{+\infty} (a_n \cos nx + b_n \sin nx)$$ $$a_0 = \frac{1}{\pi} \int_0^{2\pi} x \, dx = \frac{1}{\pi} \frac{x^2}{2} \int_0^{2\pi} = \frac{1}{2\pi} 4\pi^2 = 2\pi$$ $$a_n = \frac{1}{\pi} \int_0^{2\pi} x \cos(nx) = \frac{1}{\pi} \frac{\cos(2\pi n) - 1 + 2\pi n \sin(2\pi n)}{n^2} = 0$$ $$b_n = \frac{1}{\pi} \int_0^{2\pi} x \sin(nx) = -\frac{1}{\pi} \frac{2\pi n \cos(2\pi n) + \sin(2\pi n)}{n^2} = -\frac{2}{n}$$ $$f(x) = \pi - \sum_{n=1}^{+\infty} \frac{2}{n} \sin nx$$