Fourier Series Calculator

Enter a function:

Variable:

From (If you want −π, enter -pi):

To (If you want −π, enter -pi):

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what's Fourier series?

"The enlargement of the periodic characteristic in phrases of infinite sums of sines and cosines is referred to as Fourier series."

Fourier collection formula:

check the given method that indicates the periodic feature f(x) in the c programming language $-L\le \:x\le \:L\:$

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

Where ;

$$ a_0=\frac{1}{2L}\cdot \int _{-L}^Lf\left(x\right)dx $$

$$ a_n=\frac{1}{L}\cdot \int _{-L}^Lf\left(x\right)\cos \left(\frac{n\pi x}{L}\right)dx,\:\quad \:n>0 $$

$$ b_n=\frac{1}{L}\cdot \int _{-L}^Lf\left(x\right)\sin \left(\frac{n\pi x}{L}\right)dx,\:\quad \:n>0 $$

With the assist of the Fourier coefficients calculator, you could without problems locate values towards those coefficients.

Example

Calculate the Fourier collection of the characteristic given underneath:

$$ f(x) = x^2 \text{ for } -\pi \leq x \leq \pi $$

Solution:

The function is:

$$ f(x) = x^2 $$

First, we determine the coefficients $a_0$, $a_n$, and $b_n$:

Step 1: Calculate $a_0$

The formula for $a_0$ is:

$$ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx $$

Substitute $f(x) = x^2$:

$$ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \, dx $$

Since $x^2$ is even, we can simplify:

$$ a_0 = \frac{2}{\pi} \int_{0}^{\pi} x^2 \, dx $$

$$ a_0 = \frac{2}{\pi} \cdot \left[ \frac{x^3}{3} \right]_0^\pi $$

$$ a_0 = \frac{2}{\pi} \cdot \frac{\pi^3}{3} = \frac{2\pi^2}{3} $$

Step 2: Calculate $a_n$

The formula for $a_n$ is:

$$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx $$

Substitute $f(x) = x^2$:

$$ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x^2 \cos(nx) \, dx $$

Since $x^2 \cos(nx)$ is even:

$$ a_n = \frac{2}{\pi} \int_{0}^{\pi} x^2 \cos(nx) \, dx $$

Using integration by parts (or an online integral calculator):

$$ a_n = \frac{4(-1)^n}{n^2} $$

Step 3: Calculate $b_n$

The formula for $b_n$ is:

$$ b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin(nx) \, dx $$

Since $x^2 \sin(nx)$ is odd:

$$ b_n = 0 $$

Final Fourier Series

Combining the results, the Fourier series is:

$$ f(x) = \frac{2\pi^2}{3} + \sum_{n=1}^\infty \frac{4(-1)^n}{n^2} \cos(nx) $$

Even here, a Fourier series calculator can help simplify such calculations effectively.

How Fourier series Calculator Works?

Every time you come across complicated capabilities, our unfastened online fourier series calculator is here that will help you out in determining accurate outcomes. you may get a proper situation of the calculations by means of the usage of our calculator.

Input:

First, write your characteristic inside the drop down listing
After this, pick the variable w.r t that you need to determine the Fourier series growth
input the lower and top limits
click on ‘calculate’

Output: The Fourier growth calculator calculates: