- Half range Fourier series are used to represent functions that are periodic over only half the range of the full period.
- For an odd function, the series consists of only sine terms.
- For an even function, the series consists of only cosine terms.
- Half range Fourier series are used in solving differential equations and boundary value problems.
Important Formulas to Remember in these topic
- Sine and cosine series over the interval (0, π):
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\cos(nx) + \sum_{n=1}^{\infty} b_n\sin(nx)$$
$$a_0 = \frac{2}{\pi} \int_{0}^{\pi} f(x) dx$$
$$a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x)\cos(nx) dx$$
$$b_n = \frac{2}{\pi} \int_{0}^{\pi} f(x)\sin(nx) dx$$