## Introduction to the Fourier Transform Calculator

The **fourier series calculator** is an online application used to evaluate any variable function's Fourier coefficients. This online tool is based on the Fourier series of coefficients.

Also, dive into solid revolution calculations effortlessly using our intuitive disk calculator calculus, enabling you to compute volumes by rotating curves around the x-axis or y-axis.

## What is the Fourier Series Calculator

Fourier integral calculator is an online tool that helps us to decompose space-dependent functions into the function of time. It is used for the evaluation of real variable functions. The fourier calculator is used to **evaluate the Fourier coefficients online**. This tool also shows the graphical representation of the results of the variable function.

On the other hand, if you want to solve complex functions of differential equation, the laplace transformation calculator will provide assistance in that type of transformation online.

## Formula Used by Fourier Sine Transform Calculator

$$ \mathcal F \{f(x)\} \;=\; F(k) \;=\; \frac{1}{\sqrt{2 \pi}} \int_{-∞}^{∞} e^{ikx} f(x) dx $$

## How to Compute Fourier in Fourier Calculator?

It is quite easy to to transform the fourier integral in this fourier transform calculator with steps. This online calculator uses the following tools to **calculate fourier transform online**:

**Example: Find the Fourier transform of exp(-ax ^{2})**

Given that, We have to prove:

$$ F(k) \;=\; \mathcal F \{exp(-ax^2)\} \;=\; \frac{1}{\sqrt{2a}} exp - \frac{k^2}{4a} \;\;\;\;\;\;\;\; ,a > 0 $$

Here we have, by definition

$$ F(k) \;=\; \frac{1}{\sqrt{2 \pi}} \int_{-∞}^{∞} e^{ikx - ax^2} dx $$ $$ =\; \frac{1}{\sqrt{2 \pi}} \int_{-∞}^{∞} exp \Biggr[ -a \left(x + \frac{ik}{2a} \right)^2 - \frac{k^2}{4a} \Biggr] dx $$ $$ =\; \frac{1}{\sqrt{2 \pi}} exp(\frac{-k^2}{4a}) \int_{-∞}^{∞} e^{-ay^2} dy \;=\; \frac{1}{\sqrt{2 \pi}} exp \left( \frac{-k^2}{4a} \right) $$

In which the cange of variable **y = x + (ik/2a)** is used. The above result is correct, but the change of variable may verify by using the coplex analyis method. if a = 1/2

$$ \mathcal F \{ e^{ \frac{-x^2}{2} } \} \;=\; e^{ \frac{-k^2}{2} } $$

This shows the fourier tansform of the given function.

**Related:** Explore various mathematical tools on our website, including the volume of a washer calculator and the volumes by cylindrical shells calculator, to enhance your understanding and computation skills.

## How does Fourier Transform Calculator with Steps Works?

The **Fourier Series Calculator** allows the user to enter piecewise functions, which are defined as up to 5 pieces.

**Input**

Some examples are

if f(x) = e^{3x} → enter e^3x

if f (x, y) = sine^{3x} -> enter sin(e^(3*x))

- First of all, select the number of coefficients of the variable function to calculate and enter them into the Coefficient number box.
- Secondly, enter the lower integration limit in the given field.
- Thirdly, enter the upper integration limit (the total range) in the required field.
- Then enter the function of the real variable such as x.
- And if the function is in chunks, enter the upper limit of the first interval in the required field and add the function from that point to the next interval.
- And if the function is in chunks, enter the upper limit of the first interval in the required field and add the function from that point to the next interval.
- And if there are more chunks, then enter the upper end of the next sub-interval in the required field, and enter the function from that point to the next interval.
- If there are more chunk repeats in the previous step in the given fields, then Fourier Series Calculator lets you add up to 4 sub-intervals.

**Output:**

After a few seconds, a new window opens showing you the A_{n} and A_{n} Fourier Series coefficients for the function which is given, also it will show you some statistical and graphical representation of the solution.

Now click on results, if you want to see the graphical representation of the function and the previously calculated Fourier series.

Now enter the "df(x)/dx" to get the analytical results of the derivative of the variable function.

Now add the Integrals [a, b]" to get the initials of the function in the interval introduced.

Utilize this calculator to analyze periodic functions, and enhance your numerical analysis skills by exploring related techniques such as the trapezoidal sum calculator and simpson's rule error calculator for accurate numerical integration.

## How to Find Fourier Integral Calculator?

To find the Fourier Cosine transform calculator or the Fourier sine transform calculator for the **calculation of coefficients of the Fourier series**, just follow up the following steps:

- First of all, open your default browser's home screen.
- Now, enter the calculator's keyword, that is Fourier integral calculator in the search bar.
- Now wait for your searched results.
- Now select the Integral Calculator from Google suggestions.
- And you will get, the Fourier transform online calculator.

After opening this tool from the site, now click on the fourier transform calculator with steps for the evaluation of your problem. Now, simply added the values in the required fields to get the results.

## Benefits of Using Fourier Series Calculator with Steps

The Fourier integral calculator with steps has amazing benefits for the users and the students. It gives you accurate guidance in solving the coefficients of the given variable function from the **Fourier series**. This fourier cosine transform calculator is free of cost and is available online. It gives users free services without any subscription charges and provides accurate results. The **Fourier transform calculator** with steps has the following benefits:

- The Fourier series calculator with steps is a time-saving tool.
- This calculator can evaluate the different functions' limits or sin/cos values.
- It helps to find the definite integrals of time function in terms of frequency.
- The Fourier integral calculator is a reliable tool.
- It gives you accurate results.
- This tool is fast and easy to use.
- The Fourier Cosine transform calculator or the Fourier sine transform calculator has a friendly interface among its users.

In conclusion, our calculator provides a powerful tool for analyzing periodic functions and understanding their harmonic components.

Take your mathematical exploration further by complementing your knowledge with our riemann integral calculator, enabling you to approximate definite integrals using various summing methods. Dive deeper into the fascinating world of mathematical analysis and numerical computations with our suite of interactive calculators.