Fourier transform of a Gaussian

In summary, the conversation is about finding the Fourier transform of a Gaussian function. The attempt at a solution involves breaking the exponentials into sine and cosine terms, which results in an integral that can be evaluated using a standard gamma function. The second part of the integral is a constant exponential. The correct solution involves combining the exponentials, completing the square in x, and doing a change of variables. The result is a constant times a Gaussian function in k.
  • #1
Kolahal Bhattacharya
135
1

Homework Statement



I need to have the Fourier transform of a Gaussian

Homework Equations





The Attempt at a Solution



∫(exp[-ax^2])(exp[-ikπx]) dx

I tried by braking the last exponential into sine and cosine terms.The sine term is odd and it cancels.Then,I cannot evaluate the remaining part.Please help.
 
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  • #2
Don't do that. Combine the exponentials, complete the square in x and do a change of variables.
 
  • #3
OK,what I am getting is a standard gamma function type of integral(that I can find) and the 2nd part is an ordinary constant exponential.
So,is this what you meant?I hope there is no more subtlity in this problem.
 
  • #4
You should be getting that the Fourier transform of a gaussian in x is a constant times a gaussian in k. No, it's not subtle.
 

1. What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a signal or function into its constituent frequencies. It is used in various fields, including mathematics, physics, engineering, and signal processing.

2. What is a Gaussian function?

A Gaussian function, also known as a normal distribution, is a type of probability distribution that is commonly used to model natural phenomena. It is characterized by a bell-shaped curve and is often used to represent the distribution of data in a population.

3. How is a Gaussian function related to a Fourier transform?

The Fourier transform of a Gaussian function is another Gaussian function, with the width of the Gaussian in the spatial domain being inversely proportional to the width of the Gaussian in the frequency domain. In other words, a narrower Gaussian in the spatial domain corresponds to a wider Gaussian in the frequency domain, and vice versa.

4. What is the importance of the Fourier transform of a Gaussian?

The Fourier transform of a Gaussian is important because it allows us to analyze signals and functions in the frequency domain. This can be useful in many applications, such as image processing, data compression, and filtering. The Gaussian function is also a fundamental building block in many mathematical and scientific models, so understanding its Fourier transform is crucial in these fields.

5. How is the Fourier transform of a Gaussian calculated?

The Fourier transform of a Gaussian function can be calculated using a mathematical formula or by using software such as MATLAB. The formula involves integrating the Gaussian function with a complex exponential function, and the result is a new function in the frequency domain. The exact method of calculation may vary depending on the specific parameters of the Gaussian function.

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