Fourier Transforms: Solving Homework Problem 1/(1+t^2)

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In summary: Since the function w(t)=1/(1+t^2) is not in the form of e-a|t|, the Fourier Transform cannot be directly applied. You can use the duality principle to rewrite the function in that form, and then use the Fourier Transform relationship to find the transform. However, this method can be quite complex and may require a different approach. It may be more efficient to use a table of Fourier transforms to find the transform of w(t).
  • #1
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Homework Statement



I am asked to find the Fourier transform of w(t)=1/(1+t^2)

Homework Equations


I know the only equations I need are:

attachment.php?attachmentid=40850&stc=1&d=1320995015.png


The Attempt at a Solution



Therefore, I attempted the following:

attachment.php?attachmentid=40851&stc=1&d=1320995183.png


I tried to expand the function in the integral and got:

attachment.php?attachmentid=40852&stc=1&d=1320995303.png


I then tried to solve for this integral in terms of t from -1/2 to 1/2, and instead, the calculator says "Non-real result" and doesn't give me an answer. I know that exp(-ift)=cos(ft)+isin(ft) so the equation contains a complex part. But is there something else I need to do that I missed in my process? I even tried entering this integral into MatLab and still get the same answer as my Ti-89. I am now so frustrated.

Can someone please help me? I don't want to do a fft on Matlab or anything. I need to figure out this method. My assignment is due soon, so any help is greatly greatly appreciated. Thank you.
 

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  • #2
Your problem will be that a fractional power of (-1) is not defined in the real number domain.
You get a complex result.
 
Last edited:
  • #3
I like Serena said:
Your problem will be that a fractional power of (-1) is not defined in the real number domain.
You get a complex result.

I'm going from the time domain to the frequency domain, so the exponential must be negative right?
But still, I tried removing the negative sign... and the result remains the same. I get no answer, and the integral just stays as is. Any other ideas on what could be wrong? I'm using a Ti-89 calculator.
 
  • #4
The exponential would still take on fractional values for an f between -(1/2) and +(1/2).

But either way, this integral is hard to evaluate and it does not become easier by changing the limits of the integral.
Can it be that you're supposed to look it up in a table of Fourier transforms?
 
  • #5
I don't know why you have the limits of integration as -1/2 and 1/2. Shouldn't they be -∞ and ∞?

Anyway, I was able to get the Fourier transform of 1/(1+t^2) using the Fourier Transform relationship for e-a|t|, and using the duality principle.
 
  • #6
There is a more productive way to write

[tex]e^{-i2\pi ft}.[/tex]
 

1. What is a Fourier transform?

A Fourier transform is a mathematical tool that is used to decompose a function into its individual frequency components. It essentially converts a function from its original domain (e.g. time) to its frequency domain.

2. How do Fourier transforms work?

Fourier transforms work by breaking down a function into an infinite sum of sine and cosine functions with different frequencies. This allows us to analyze the frequency components of a function and understand how it behaves.

3. What is the purpose of solving homework problem 1/(1+t^2) with Fourier transforms?

The purpose of solving this homework problem with Fourier transforms is to demonstrate the usefulness and applicability of this mathematical tool in solving real-world problems. It also helps to develop a deeper understanding of Fourier transforms and their applications.

4. How do you solve the homework problem 1/(1+t^2) using Fourier transforms?

The solution involves using the Fourier transform formula, which converts the function into its frequency domain representation. By manipulating the formula and applying some algebraic techniques, the solution can be obtained.

5. What are the applications of Fourier transforms?

Fourier transforms have a wide range of applications in various fields such as signal processing, image and sound analysis, data compression, and solving differential equations. They are also used in physics, engineering, and other sciences to analyze and understand complex systems.

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