# Fourier Transform Calculator

Want to perform online fourier transform? Get the assistance of our amazing fourier transform calculator step by-step in no time.

## Introduction to the Fourier Transform Calculator

The fourier transform graph 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 Fourier Transform?

The Fourier transform is a fundamental transformation used to analyze functions or signals in terms of their frequency components. It decomposes a function into its constituent frequencies, revealing the underlying frequency spectrum. This transformation is invaluable in various fields such as signal processing, image analysis, quantum mechanics, and many others.

There are two types of Fourier transform: the continuous Fourier transform (CFT) and the discrete Fourier transform (DFT). The CFT deals with continuous signals defined over an infinite interval, while the DFT is used for discrete signals sampled at specific points in time or space.

## What is the Fourier Transformation 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

Imagine there is a complex signal that represents a sound wave or an image. You can break down it into simpler sinusoidal waves of different frequencies and amplitudes. The fourier transformation online tool will allows us to precisely identify and quantify these individual frequency components.

Mathematically, the fourier analysis calculator uses following formula to perform online fourier transform of a function.

$$F(ω) \;=\; \int_{−∞}^{∞} f(t)e^{−iωt} dt$$

Here,

F(ω) represents the transformed function in terms of frequency ω

f(t) is the original function in the time domain.

## How to Calculate Fourier Transform?

Calculating the Fourier transform involves several steps and methods, depending on whether you're dealing with continuous or discrete signals. But the major steps that you must follow even in both types are:

• Understand the function or signal you want to analyze. Make sure it is a integrable over its domain for Fourier transform.
• Use the Fourier transform formula based on whether the signal is continuous or discrete one.
• Compute the Fourier transform coefficients based on the respective formulas.
• Interpret the results in terms of the frequency spectrum of the original function. Identify the dominant frequencies and their corresponding magnitudes.

Thus, here after that you will get the foruier transofrm of the integrand function.

## How to Compute Fourier in Fourier Calculator?

It is quite easy to to transform the fourier integral in this fourier transform calculator with steps. It is a totally AI based calculator that analyze nature of your question problem and provide solution by thinking about it like a human mind.

This online calculator uses the above manual procedure to calculate fourier transform online. Let's understand it with some examples, how to do fourier analysis online in no time.

### Solved Examples of Calculator Fourier Transform

Example: Find the Fourier transform of exp(-ax2)

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 transformation calculator allows the user to enter piecewise functions, which are defined as up to 5 pieces.

Input

Some examples for online fourier transform are
if f(x) = e3x → enter e3x
if f (x, y) = sine3x -> enter sin(e3x)

• 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 coefficients calculator lets you add up to 4 sub-intervals.

Output:

After a few seconds, a new window opens showing you the An and An 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 "calculator fourier transform","fourier transformation online" or "fourier analysis online" 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 step by-step for the evaluation of your problem. Now, simply added the values in the required fields to get the results.

## Benefits of Using Fourier Coefficients Calculator

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 transform solver 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 transform graph calculator is a time-saving tool.
• It 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 analysis 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, Fourier transform solver 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.

### What does a Fourier transform do?

The Fourier transform is a function that splits a waveform, which is a function of time, into the pieces i.e., frequencies that build it up. The result generates is a complex-valued function of frequency.

### Find the Fourier Transform of f(x)=e^(x)

To find the Fourier transform of f(x)=e^(x) the Fourier transforms calculator analyze the integral,

• As f(x) = ex, the function leads to infinity as x increases so, the Fourier transform diverges depending on the integration limits. This divergence can be avoided by considering the area where x is finite
• The given integral will become,
• $$F(k) \;=\; \int_{-\infty}^{\infty} e^x e^{ikx} dx \;=\; \int_{-infty}^{\infty} e^{(1-ik)^x} dx$$
• For an integral to converge the real part of the exponent must be negative which will ensure that the integrals converge as x tends to ∞.
• Evaluate integral in the limits of convergence,

$$F(k) \;=\; \int_{0}^{\infty} e^{(1-ik)x} dx$$

$$F(k) \;=\; \frac{1}{1-ik}$$

### Show that Fourier Transform of e^-x^2/2 is Self Reciprocal

To show that the Fourier transform of f(x) =e^-x^2/2 is self reciprocal it is important to find the Fourier transform of the function using the Fourier transform finder which uses the given formula,

$$F(k) \int_{-\infty}^{\infty} f(x) e^{ikx} dx$$

Putting the value in this formula,

$$F(k) \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}} e^{-ikx} dx$$

Complete the square,

$$-\frac{x^2}{2} - ikx \;=\; -\frac{x^2}{2} -ikx \;=\; -\frac{1}{2} \biggr(x^2 + 2ikx \biggr)$$

$$-\frac{1}{2} (x + ik)^2 + \frac{k^2}{2}$$

Rewriting the integral,

$$F(k) \;=\; e^{-\frac{k^2}{2}} \int_{-\infty}^{\infty} e^{-\frac{1}{2}(x+ik)^2} dx$$

Gaussian integral for this is,

$$\int_{-\infty}^{\infty} e^{-\frac{x^2}{2}} dx \;=\; \sqrt{2 \pi}$$

So the simplified expression is,

$$\int_{-\infty}^{\infty} e^{-\frac{1}{2}(x+ik)^2} dx \;=\; \sqrt{2 \pi}$$

Putting the values,

$$F(k) \;=\; e^{-\frac{k^2}{2}} \sqrt{2 \pi}$$

The Fourier transform would be,

$$G(x) \;=\; \int_{-\infty}^{\infty} \sqrt{2 \pi e} -\frac{k^2}{2} e^{-ikx} dk$$

### Find Fourier sine Transform of e^-3x

The Fourier sine transform of the function is given as,

$$F_s (k) \;=\; \int_{0}^{\infty} f(x) sin(kx) dx$$

Putting the values,

$$F_s(k) \;=\; \int_{0}^{\infty} e^{-3x} sin(kx) dx$$
The integral of the product of sine function and exponential function is,

$$\int_{0}^{\infty} e^{-ax} sin(bx) dx \;=\; \frac{b}{a^2 + b^2}$$

$$F_s(k) \;=\; \frac{k}{3^2 + k^2} \;=\; \frac{k}{9 + k^2}$$

So, the result is,

$$F_s(k) \;=\; \frac{k}{9+k^2}$$

### Find the Fourier Series of f(x)=1-x^2 in (-1 1)

To find the Fourier series of f(x) = 1 - x2 in the integral, the Fourier transform generator uses the general formula for the Fourier series,

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

Where the coefficients are,

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

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

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

Thus, the fourier series is,

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

Finding a0:

$$a_0 \;=\; \frac{1}{2} \int_{-1}^{1}(1 - x^2)dx \;=\; \frac{1}{2} \biggr( \biggr[ x - \frac{x^3}{3} \biggr]_{-1}^{1} \biggr)$$

Putting the values,

$$a_0 \;=\; \frac{1}{2} \biggr( 1 - \frac{1}{3} - \biggr(-1 + \frac{1}{3} \biggr) \biggr) \;=\; \frac{1}{2} \times \frac{4}{3} \;=\; \frac{2}{3}$$

Finding an:

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

Splitting in two parts:

$$\int_{-1}^{1} cos(\pi n x) dx \;=\; 0$$

$$\int_{-1}^{1} x^2 cos(\pi n x) dx$$

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

Finding bn:

As f(x) is symmetric so the bn coefficients will be zero,

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

### Find the Fourier Series of f(x)=x^2 in (0 2pi)

To find the Fourier series of f(x) = x2 in the interval, use the general formula of Fourier series,

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

Where the coefficients are,

$$a_0 \;=\; \frac{1}{L} \int_0^L f(x) dx$$

$$a_n \;=\; \frac{2}{L} \int_0^{L} f(x) cos(nx) dx$$

$$b_n \;=\; \frac{2}{L} f(x) sin(nx) dx$$

Finding a0:
$$a_0 \;=\; \frac{1}{2 \pi} \int_{0}^{2 \pi} x^2 dx \;=\; \frac{1}{2 \pi} . \frac{x^3}{3} \biggr|_{0}^{2\pi} \;=\; \frac{1}{2\pi} . \frac{(2 \pi)^3}{3} \;=\; \frac{4 \pi^2}{3}$$

Finding an:

$$a_n \;=\; \frac{1}{\pi} \int_{0}^{2 \pi} x^2 cos(nx) dx$$

Using integration by parts with u = x2 and dv = cos(nx)dx, we get

$$du \;=\; 2x\;dx \;and\; v \;=\; \frac{1}{n} sin(nx)$$

So, it becomes,

$$\frac{x^2}{n} sin(nx) - \int \frac{2x}{n} sin(nx) dx$$

$$\int_0^{2 \pi} x^2 cos(nx) dx \;=\; \frac{2 \pi^2}{n^2}$$

Thus,

$$a_n \;=\; \frac{1}{\pi} . \frac{2 \pi^2}{n^2} \;=\; \frac{4 \pi}{n^2}$$

Finding bn:

$$b_n \;=\; \frac{2}{2 \pi} \int_{0}^{2 \pi} x^2 sin(nx)dx \;=\; 0$$

Therefore, the results are:

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

### Is There a Fourier Series in all Functions?

No, the Fourier series can only represent periodic functions. The Fourier series can not be there in all mathematical functions.

### Why is the Fourier transform Useful?

The Fourier transforms gives your insight into what sine wave frequencies make up a signal. You can apply knowledge of the frequency domain from the Fourier transform in very beneficial methods. Likewise: Audio processing, detects specific tones or frequencies, and even alters them to produce a new signal.