## Introduction to Fourier Series Calculator

Fourier series calculator is an advanced calculus tool that helps to calculate the Fourier series of the periodic function. It determines the infinite sum of complex variables of periodic functions that expand in terms of cosine and sine series.

Our Fourier sine series calculator is very helpful for students, engineers, and researchers who do not want to be involved in complex and lengthy calculations to solve the Fourier series.

That is why we designed the Fourier cosine series calculator because it enables you to solve various types of Fourier series questions in a fraction of a second.

This calculator is used when working with functions that repeat over a fixed interval, such as sine waves, square waves, or sawtooth waves. Additionally, if you would like to be t convert a non-periodic function into a periodic function, you can go through our online fourier transform. Our calculator also solves the Fourier complex function with undefined limits in the real value function.

## What is the Fourier Series?

Fourier series is defined as the expansion of infinite sums of sine and cosine series in periodic functions. It changes complex variables into periodic functions because its series expands in terms of periodic functions only not for frequency functions.

The Fourier Series allows us to decompose complex periodic functions into simpler sinusoidal components, providing insights into their frequency content and behavior. It has widespread applications in various fields, including signal processing, engineering, physics, and applied mathematics.

If you want to explore further into Fourier Series or related mathematical concepts, you might also be interested in learning about the Laplace ransform. You can use our laplace transform calculator with steps. This calculator helps users perform Laplace Transformations efficiently, facilitating the analysis of functions and systems in the s-domain.

## Formula Behind Fourier Sine Series Calculator

Fourier series expansion of sine and cosine represents in terms of f(x). The Fourier series calculator uses the following formula,

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

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

$$ a_n \;=\; \frac{1}{L} . \int_{-L}^{L} f(x) cos \biggr( \frac{n \pi x}{L} \biggr) dx, \; \; \; \; n \gt 0 $$

$$ b_n \;=\; \frac{1}{L} . \int_{-L}^{L} f(x) sin \biggr( \frac{n \pi x}{L} \biggr) dx, \; \; \; \; n \gt 0 $$

Where,

- a0: constant terms coefficient
- an: represents the cosine Fourier series coefficient
- bn: represent sine Fourier series coefficient
- L and -L: the upper and the lower limits
- nx/L: the cosine and sine variables

## Working of Fourier Cosine Series Calculator

Fourier series coefficients calculator has a simple working method to evaluate a Fourier series for sine, cosine, or simple Fourier series function. Its server has all Fourier series formulas that enable you to solve complicated Fourier series solutions.

When you add the Fourier function in this Fourier series coefficient calculator, it will analyze the given function whether it belongs to the simple Fourier, cosine, or sine Fourier series. After recognizing the function types it applies the Fourier series formula. If the function type is a simple Fourier series then an or bn is zero and a0 gives the result.

If the series type in our Fourier series expansion calculator is cosine series then an gives the solution but bn is zero. Similarly, if the Fourier series is a sine series then an is zero but bn gives the solution of the sine Fourier series. These are solved by the integration method and then the upper and lower limits are applied.

After getting the values add a0, an, or bn value in f(x) and give the solution of the given function. For further exploration of signal processing techniques, such as the Z-transform, you can utilize our Z Transform Calculator, providing deeper insight into signal analysis and processing.

Now let us examine an example of a Fourier series in the Fourier series piecewise calculator to understand its working process.

## Solved Example of Fourier Series

The Fourier series calculator is always here to help but it's also important to know the manual calculation of the process. So we are going to give you an example to let you know about the manual calculations.

### Example:

Determine the Fourier series of the function f(x) = 1 - x^{2} in the interval -1,1?

**Solution:**

The Fourier series of the function f(x) in the interval,

-L, L i.e. -L ≤ x ≤ L is written as:

$$ f(x) \;=\; A_0 + \sum_{n=1}^{\infty} A_n . cos \biggr( \frac{n \pi x}{L} \biggr) + \sum_{n=1}^{\infty} B_n . sin \biggr( \frac{n \pi x}{L} \biggr) $$

Here,

$$ A_0 \;=\; \frac{1}{2L} . \int_{-L}^{L} f(x) dx $$

$$ A_n \;=\; \frac{1}{1} . \int_{-1}^{1} f(x) cos \biggr( \frac{n \pi x}{1} \biggr) dx, n \gt 0 $$

$$ B_n \;=\; \frac{1}{1} . \int_{-1}^{1} f(x) sin \biggr( \frac{n \pi x}{1} \biggr) dx, n \gt 0 $$

By applying the formula:

$$ f(x) \;=\; \frac{1}{2.1} . \int_{-1}^{1} (1-x^2)dx + \sum_{n=1}^{\infty} \frac{1}{1} . \int_{-1}^{1} (1-x^2) cos \biggr(\frac{n \pi x}{1} \biggr) dx . cos \biggr(\frac{n \pi x}{1} dx + \sum_{n=1}^{\infty} $$

$$ \int_{-1}^{1} (1-x^2) sin \biggr( \frac{n \pi x}{1} \biggr) dx. sin \biggr( \frac{n \pi x}{1} \biggr) dx $$

Simplifying the definite integral,

$$ \frac{1}{2.1} \biggr( \frac{4}{3} \biggr) + \sum_{n=1}^{\infty} \frac{1}{1} \biggr( - \frac{4(-1)^n}{\pi^2 n^2} \biggr) cos \biggr( \frac{n \pi x}{1} \biggr) + \sum_{n=1}^{\infty} \frac{1}{1} . O. sin \biggr( \frac{n \pi x}{1} \biggr) $$

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

Thus it is the final solution of our function with specific limits. Solve this example further, you can apply the inverse Laplace transform. This will convert the given expression from the Laplace domain back to the time domain. You can use our laplace inverse transform calculator to perform this conversion efficiently.

## How to Evaluate in Fourier Series Calculator

Fourier series coefficient calculator evaluates the Fourier series problem in an easy way, you do not need to make an extra effort in the evaluation of complicated questions of the Fourier series. For this, you need to follow our instructions while using it. These instruction are

Additionally, if you're dealing with differential equations involving Laplacian operators, you should use our laplacian operator mask calculator . It can assist you in solving differential equations involving Laplacian operators efficiently, complementing your Fourier series analysis.

- Enter the Fourier series function in the respective field of Fourier series expansion calculator.
- Choose the variable of integral from the given list
- Add the upper and lower limits for the Fourier series solution
- Select the type of series (Sine, cosine, or general form of Fourier series).
- Choose the order of expansion from the given list
- Click on the calculate button to get the solution of the Fourier series
- Click on the Recalculate button for more calculations of the Fourier series.

## Output from Fourier Series Coefficients Calculator

You can get the solution of the Fourier series within a minute when you enter the function in the Fourier series calculator. It may include as

**Result option of the Fourier series**

This option provides you with a solution for the Fourier series

**Possible steps of the fourier series**

It gives you a complete step-by-step process of the Fourier series

**Graph plot option of fourier series**

Plot option draw a graph according to the given Fourier series solution

Moreover, if you're dealing with integrals related to your Fourier series calculations, you should use our solve integration online. It can assist you in computing integrals efficiently, complementing your Fourier series analysis.

## Benefits of Fourier Series Coefficient Calculator:

Fourier sine series calculator provides you with tons of benefits whenever you use it to calculate the Fourier series for sine or cosine series expansion. These benefits are given below.

If you're interested in exploring more mathematical tools, you might also find our all calculator useful for accessing a wide range of calculators for various mathematical operations and analyses.

- Fourier series saves time that you consume while doing lengthy calculations
- It can solve various types of complex Fourier series solutions.
- Our fourier cosine series calculator can operate through a computer, laptop, or mobile for the Fourier series.
- The fourier series coefficients calculator is used in many fields like physics (thermodynamics), IT field telecommunications, data processing, voice signal processing, etc.
- This Fourier series calculator has a user-friendly interface so that you can solve it easily.