Fourier Series Classroom

101.1 Introduction to Fourier Series

Fourier Series provide a way to represent a periodic function as an infinite sum of sines and cosines. This concept is fundamental in many areas of science and engineering, particularly in signal processing, image compression, and solving partial differential equations.

The core idea is that any "well-behaved" periodic function can be decomposed into a unique set of sinusoidal components, each with a specific amplitude and phase. This allows us to analyze complex periodic phenomena by breaking them down into simpler, oscillating parts.

101.2 Periodic Functions

A function $f(x)$ is said to be periodic with period $T$ if $f(x+T) = f(x)$ for all $x$ in the domain of $f$, where $T$ is a positive constant. The smallest positive value of $T$ for which this holds is called the fundamental period.

For Fourier Series, we often work with functions having a period of $2\pi$. Examples of periodic functions include $\sin(x)$ and $\cos(x)$, both of which have a fundamental period of $2\pi$.

101.3 Fourier Series Definition (Period $2\pi$)

For a function $f(x)$ defined on the interval $[-\pi, \pi]$ and having a period of $2\pi$, its Fourier series is given by:

The coefficients $a_0$, $a_n$, and $b_n$ are calculated using the Euler formulas:

These integrals effectively extract the "amount" of each cosine and sine component present in the function $f(x)$.

101.4 Fourier Problems (Period $2\pi$)

Let's find the Fourier series for a simple piecewise function defined over $[-\pi, \pi]$.

102.1 Non-Periodic Function Expansion (Range $2\pi$)

While Fourier series are inherently for periodic functions, we can represent non-periodic functions over a finite interval using Fourier series by creating periodic extensions.

If a function $f(x)$ is defined only on an interval $[0, \pi]$ (or $[0, L]$ in general), we can extend it to $[-\pi, \pi]$ (or $[-L, L]$) as either an even or an odd function, and then apply the standard Fourier series formulas. This leads to:

• Fourier Cosine Series: For an even extension, $b_n = 0$, and the series only contains cosine terms. $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx)$$ $$a_0 = \frac{2}{\pi} \int_{0}^{\pi} f(x) \,dx$$ $$a_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \cos(nx) \,dx$$
• Fourier Sine Series: For an odd extension, $a_n = 0$, and the series only contains sine terms. $$f(x) = \sum_{n=1}^{\infty} b_n \sin(nx)$$ $$b_n = \frac{2}{\pi} \int_{0}^{\pi} f(x) \sin(nx) \,dx$$

The choice of extension depends on the desired properties of the series or the boundary conditions in differential equations.

102.2 Fourier Problems (Non-Periodic)

Let's find the Fourier sine series for a simple function defined on $[0, \pi]$.