Fourier Series Solver

$f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$

Fourier Series (Period 2π)
Even, Odd Functions » Harmonic Analysis » Heaviside » Differential Equations » Laplace Transforms

Fourier Series for Periodic Functions

A Fourier series is an expansion of a periodic function into a sum of sines and cosines. It allows us to represent complex periodic waveforms as a combination of simpler, harmonically related sinusoidal functions. This concept is fundamental in many fields, including signal processing, image compression, and solving partial differential equations.

Understanding Periodic Functions

A function $f(x)$ is said to be periodic with period $T$ if there exists a positive constant $T$ such that $f(x+T) = f(x)$ for all $x$ in the domain of $f$. The smallest such positive $T$ is called the fundamental period. Examples include $\sin(x)$ (period $2\pi$) and $\cos(x)$ (period $2\pi$).

$$f(x+T) = f(x)$$

Formulas for a Fourier Series and Coefficients

For a function $f(x)$ defined on an interval $-L \le x \le L$ (or any interval of length $2L$), its Fourier series is given by:

$$f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right)$$

Where $L$ is half the period, i.e., $T = 2L$. The coefficients $a_0, a_n,$ and $b_n$ are called the Fourier coefficients and are calculated using the following integral formulas:

$$a_0 = \frac{1}{2L} \int_{-L}^{L} f(x) \,dx$$ $$a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) \,dx \quad \text{for } n=1, 2, 3, \ldots$$ $$b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) \,dx \quad \text{for } n=1, 2, 3, \ldots$$

Note: Symbolic integration to find exact coefficients is complex for a web browser environment. This solver will numerically approximate the integrals to find the coefficient values, allowing for a numerical approximation of the Fourier series. For exact analytical solutions, manual calculation or specialized symbolic software is typically required.

Obtain Fourier Series for Given Functions (Numerical Approximation)

Function $f(x)$ (e.g., x):

Period $T$ (e.g., 2*pi for sin(x), 2 for x over [-1,1]):

Number of terms $N$ (e.g., 5, higher N gives better approximation):

Enter function, period, and number of terms, then calculate.

Fourier Series for Non-Periodic Functions (Over a Limited Range)

While Fourier series are fundamentally used for periodic functions, they can also represent a non-periodic function over a specific, limited interval. When we do this, we are effectively treating the non-periodic function as one period of a hypothetical periodic function, which is then called its periodic extension.

The Fourier series will accurately represent the original non-periodic function *only within its defined interval*. Outside this interval, the Fourier series will repeat the pattern defined within that interval, which may not match the original non-periodic function's behavior (if it continues outside the interval).

Fourier Series over a $2\pi$ Range (i.e., $L=\pi$)

A common scenario is to expand a non-periodic function $f(x)$ defined over the interval $[-\pi, \pi]$ (or any interval of length $2\pi$). In this case, the half-period $L = \pi$, and the Fourier series formulas simplify to:

$$f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)$$

Where the coefficients are:

$$a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) \,dx$$ $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \,dx \quad \text{for } n=1, 2, 3, \ldots$$ $$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \,dx \quad \text{for } n=1, 2, 3, \ldots$$

Note: As before, symbolic integration is not performed client-side. The integrals are numerically approximated. The plot will show the original function only in its defined range $[-\pi, \pi]$ and the periodic extension of its Fourier series outside this range to illustrate the concept of periodic extension.

Calculate Fourier Series for Non-Periodic Function over $[-\pi, \pi]$