- Fourier series represent periodic functions as a sum of sine and cosine functions.
- The coefficients of a Fourier series can be found using Euler's formulas.
- Fourier series are used in signal processing, communication systems, and other fields.
- Fourier series converge to the function they represent, under certain conditions.
Important Formulas to Remember in these topic
- Definition of Fourier series:
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \Big[ a_n\cos(nx) + b_n\sin(nx) \Big]$$
- Euler's formulae for determining Fourier coefficients:
$$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx$$
$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx) dx$$
$$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx) dx$$