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Fourier Series Calculator

Write down a periodic function in the designated field, and the calculator will compute its Fourier series, displaying the calculations.

Enter a function:

Variable:

To (If you want −π, enter -pi):

From (If you want −π, enter -pi):

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Fourier Series Calculator

Use our free Fourier Series Calculator to compute the Fourier series of any periodic function. Before using the tool, let’s review the key concepts behind Fourier series.

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.

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 #1

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:

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

Using a Fourier series calculator ensures accuracy and saves time compared to manual integration.

How the Fourier Series Calculator Works

Our tool quickly computes the series and coefficients for any periodic function.

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

References

Wikipedia: Fourier Series

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