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Fourier Series Calculator

Enter a periodic function and obtain its full Fourier series representation, including a₀, aₙ, and bₙ coefficients.

Enter a function:

Variable:

From (If you want −π, enter -pi):

To (If you want −π, enter -pi):

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Fourier Series Calculator:

This Fourier Series Calculator helps you find the Fourier series representation and Fourier coefficients of a periodic function quickly and accurately. Whether you are solving mathematics problems, studying signal processing, or working on engineering applications, the tool simplifies the calculation process. It provides the results needed to analyze periodic functions and signals.

What Is a Fourier Series?

In mathematics, a Fourier series represents a periodic function as an infinite sum of sine and cosine functions. This decomposition helps analyze signals, vibrations, and other periodic phenomena.

What is Fourier series formula?

For a function $f(x)$ defined on the interval $-L \le x \le L$, the Fourier series is written as:

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

Where the Fourier coefficients are defined by:

$$ a_0 = \frac{1}{2L} \int_{-L}^{L} f(x) \, dx $$
$$ a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n \pi x}{L}\right) dx, \quad n > 0 $$
$$ b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n \pi x}{L}\right) dx, \quad n > 0 $$

With these coefficients, you can reconstruct the periodic function as a sum of sines and cosines using the calculator.

How Is the Fourier Series Calculated?

Manually computing Fourier series can be time-consuming. Our Fourier series calculator automates the process. Let’s see an example:

Example:

Compute the Fourier series for:

$$ f(x) = L - x, \quad -L \le x \le L $$

Solution:

Check the function’s symmetry:

$$ f(-x) = L - (-x) = L + x \neq f(x), \quad \text{but } f(-x) = -f(x) + 2L $$

For simplicity, if we consider the odd component, we set $a_0 = 0$ and $a_n = 0$.

Compute $b_n$ coefficients:

$$ b_n = \frac{1}{L} \int_{-L}^{L} (L - x) \sin\left(\frac{n \pi x}{L}\right) dx $$

Evaluating the integral gives:

$$ b_n = \frac{2L(-1)^n}{n\pi}, \quad n = 1, 2, 3, \dots $$

Hence, the Fourier series becomes:

$$ f(x) = \sum_{n=1}^{\infty} \frac{2(-1)^n}{n} \sin\left(\frac{n \pi x}{L}\right) $$

How the Fourier Series Calculator Works?

With our Fourier Series Calculator, you can quickly find the Fourier series and coefficients of any periodic function. Let’s see how it works.

Input:

Enter the function
Select the variable
Provide the interval’s lower and upper limits
Click ‘Calculate’

Output:

Fourier series expansion
Values of $a_0$, $a_n$, and $b_n$
Step-by-step calculation details

Reference:

Wikipedia: Fourier Series

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