## Introduction to Fourier Series Calculator:

Fourier series calculator is a tool, used to **compute** the coefficient value of Fourier series function within the given time domain. It helps to simplify the complex and lengthy process of Fourier series periodic function.

It is the best online source which provides you solutions of complex Fourier series without taking any instructions. It has an up-to-date server that allows you to solve all types of complicated Fourier series functions problems in just one click.

## What is the Fourier Series?

Fourier series is a mathematical process used to find a periodic function in complex analysis that is the **sum of sine** and cosine functions. It was developed by French mathematician Joseph Fourier in the early 19th century.

Fourier transforms extend their application from non-periodic functions to understand the frequency and harmonic components of periodic phenomena. It is widely used in various fields like signal processing, image analysis, physics, and engineering to analyze and synthesize periodic signals.

## Formula of Fourier Series:

The given function f(x) denotes the Fourier series function with the period of time T.

$$ f(x) \;=\; \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n cos \left(\frac{2n \pi x}{T} \right) + b_n sin \left(\frac{2n \pi x}{T} \right) \right] $$

Here,

a0, an ,bn: the **coefficient of Fourier series** function.

sine or cosine: trigonometric functions of T.

## How to Calculate Fourier Series?

To **calculate the Fourier series** of a periodic function you need to find the value of a0, an and bn because these cofficient values have their own formulas. Its solution depends on continuous or discrete function. Here are the general steps for calculating the Fourier series for a continuous periodic function f(x).

**Step 1**: Identify the period T of the function f(x) and the specific interval value. This is the interval over which the function repeats itself.

**Step 2**: Compute a0 with the help of formula and add the given values in the given integral function f(x). Here L and -L are the upper and the lower limit. $$ a_0 \;=\; \frac{1}{2L} \int_{-L}^{L} f(x) dx $$

**Step 3**: Calculate the value by putting the given function value in the below formula. The value of n varies. For n = 1, 2, 3,… $$ a_n \;=\; \frac{1}{L} \int_{-L}^{L} f(x) cos \frac{n \pi x}{L} dx $$

These integrals (or sums for discrete signals) compute the coefficients of cosine and sine in the Fourier series.

**Step 4**: For bn, repeat the above same procedure of finding the value of bn with its formula that is shown below. $$ b_n \;=\; \frac{1}{L} \int_{-L}^{L} f(x) sin \frac{n \pi x}{L} dx,\; n ≥ 1 $$

**Step 5**: In the Fourier series function formula, add all the coefficient values a0, a1, and bn so that you get the solution of Fourier series periodic function. $$ F(x) \;=\; a_0 + \sum_{n=1}^{\infty} \left( a_n cos \frac{n \pi x}{L} + b_n sin \frac{n \pi x}{L} \right) $$

For those who want quick solution, we offer the fourier series calculator to quickly calculate the fourier series function coefficient within a given time domain.

## Solved Example of Fourier Series:

The **solved example** of the fourier series in steps help you to understand how this fourier sine series calculator solves problems.

### Example: Find the fourier series of following:

$$ f(x) \;=\; x^2 - x \;on\; [-2, 2] $$

Determine its sum for -2 ≤ x ≤ 2.

**Solution**:

$$ f(x) \;=\; x^2 - x $$

$$ L \;=\; 2,\; -L \;=\; -2 $$

The Fourier series function is,

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

Find the coefficient value of a0, an and bn with the help of their respective formula. For a0 value:

$$ a_0 \;=\; \frac{1}{2L} \int_{-L}^{L} f(x)\; dx $$

Put the given value and solve it.

$$ a_0 \;=\; \frac{1}{4} \int_{-2}^{2} (x^2 - x) dx $$

$$ a_0 \;=\; \frac{1}{2} \int_0^2 x^2 dx \;=\; \frac{x^3}{6} \biggr|_0^2 \;=\; \frac{4}{3} $$

For an value:

$$ a_n \;=\; \frac{1}{L} \int_{-L}^{L} f(x) cos \frac{n \pi x}{L} dx $$

Put the given value and solve it.

$$ a_n \;=\; \frac{1}{2} \int_{-2}^{2} (x^2 - x) cos \frac{n \pi x}{2}\; dx,\; n \;=\; 1,2,3,... $$

$$ a_n \;=\; \int_0^2 x^2 cos \frac{n \pi x}{2} dx \;=\; \frac{2}{n \pi} [x^2 sin \frac{n \pi x}{2} \biggr|_0^2 - 2 \int_0^2 x sin \frac{n \pi x}{2} dx] $$

$$ =\; \frac{8}{n^2 \pi^2} [x\; cos \frac{n \pi x}{2} \biggr|_0^2 - \int_0^2 cos \frac{n \pi x}{2} dx] $$

$$ =\; \frac{8}{n^2 \pi^2} \left[2 cos\; n \pi - \frac{2}{n \pi} sin \frac{n \pi x}{2} \biggr|_0^2 \right] \;=\; (-1)^n \frac{16}{n^2 \pi^2} $$

For bn value:

$$ b_n \;=\; \frac{1}{L} \int_{-L}{L} f(x) sin \frac{n \pi x}{L} dx, n ≥ 1 $$

Put the given value and solve it.

$$ b_n \;=\; \frac{1}{2} \int_{-2}^{2} (x^2 - x) sin \frac{n \pi x}{2} dx,\; n \;=\; 1,2,3,... $$

$$ b_n \;=\; -\int_0^2 xsin \frac{n \pi x}{2}dx \;=\; \frac{2}{n \pi} \left[ x\; cos \frac{n \pi x}{2} \biggr|_0^2 - \int_0^2 cos \frac{n \pi x}{2} dx \right] $$

$$=\; \frac{2}{n \pi} \left[ 2 cos n \pi - \frac{2}{n \pi} sin \frac{n \pi x}{2} \biggr|_0^2 \right] \;=\; (-1)^n \frac{4}{n \pi} $$

Add the value of a0, an, and bn in the Fourier series formula and you get the solution of the Fourier series function.

$$ F(x) \;=\; \frac{4}{3} + \frac{16}{\pi^2} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} cos \frac{n \pi x}{2} + \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{n} sin \frac{n \pi x}{2} $$

## How to Use the Fourier Cosine Series Calculator?

The Fourier series coefficients calculator has a simple design that makes it easy for you to evaluate the Fourier series problems. Follow our instructions which are given as:

- Enter the given Fourier series function that you want to evaluate in the given input field.
- Check the given complex Fourier series function before clicking the calculate button of Fourier series coefficient calculator.
- Click the “
**Calculate**” button to get the result of your given Fourier series function problem. - If you are trying our Fourier series expansion calculator for the first time then you should use the load example to learn more about this concept.
- Click on the “Recalculate” button to get a new page for finding more solutions of Fourier series problems.

## Output of Fourier Series Piecewise Calculator:

The fourier series calculator gives you the **solution** from a given function when you give it an input. The results contain as:

**Result Option**:

When you click on the result option, it gives you a solution of the Fourier series function.

**Possible Steps**:

When you click on it, this option will provide you with step by step solution of Fourier series.

**Plot Option**:

It will give you a graphical representation of Fourier series function solutions to clearly understand the whole solution.

## Advantages of Fourier Sine Series Calculator:

The Fourier cosine series calculator provides you with tons of benefits that help you to calculate the fourier series of given function problems and give you solutions without any trouble. These useful features are:

- Fourier series coefficients calculator is a
**free-of-cost tool**so you can use it for free to find fourier series solutions without spending. - It gives you conceptual clarity for the fourier series process when you use it for practicing more examples.
- The fourier series coefficient calculator saves the time and effort that you consume on the calculation of complex functions for finding the fourier series problems manually.
- It is an adaptable tool that gives you precise solutions whenever you use it to calculate fourier series function without any mistakes in calculation.
- Fourier series expansion calculator enables you to use it multiple times for the evaluation of fourier series problems.
- Fourier series piecewise calculator is a handy tool because you can access it through an online platform from anywhere.