Fourier Intergrals and transforms

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In summary, the conversation discusses the process of evaluating a Fourier integral and the confusion surrounding it. The individual initially attempts to use integration by parts, but after further research realizes that it may not be applicable. Another individual suggests using a substitution and the original individual agrees, having done a similar integral before. They also mention being surprised at how simple the integral ended up being, despite its name.
  • #1
Brewer
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How do I do a Fourier integral (and what's the point of them??)

I've been asked to evaluate

[tex]F(\omega) = \frac{1}{\sqrt{2\pi}}\int dte^{-\alpha t}cos\omega t[/tex]

and I've not the foggiest idea what to do. I thought I could just go about doing in the integral by parts (limits are 0 and infinity by the way), but on further research I don't think I can do that can I?
 
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  • #2
Try integration by parts.
 
  • #3
cos(u)=(exp(iu)+exp(-iu))/2

Substitute into your integral and you will have the sum of two exp integrals. I presume you can do that.
 
  • #4
Yes that's familliar. I have done this integral before - just never dawned on me to use the substitution.

I was also a little confused when it called it a Fourier Integral. I thought it was going to be a lot more complex than it appears to be now.
 

Related to Fourier Intergrals and transforms

1. What is a Fourier Integral and transform?

A Fourier Integral and transform is a mathematical tool used to decompose a complex function into simpler components. It converts a function from the time or spatial domain to the frequency domain, allowing for analysis and manipulation of the different components of the function.

2. What is the difference between a Fourier Integral and a Fourier transform?

A Fourier Integral is used for continuous functions, while a Fourier transform is used for discrete functions. A Fourier Integral is represented by a continuous integral, while a Fourier transform is represented by a discrete sum.

3. What are the applications of Fourier Integrals and transforms?

Fourier Integrals and transforms are commonly used in signal processing, image processing, and acoustics. They are also used in fields such as physics, engineering, and mathematics for solving differential equations and analyzing data.

4. What is the relationship between Fourier transforms and Laplace transforms?

There is a strong connection between Fourier transforms and Laplace transforms. Laplace transforms are essentially an extension of Fourier transforms to functions that are not periodic. However, Laplace transforms also take into account the initial conditions of a system, while Fourier transforms do not.

5. Are there any limitations to using Fourier Integrals and transforms?

Fourier Integrals and transforms are not suitable for all types of functions. They are most effective for functions that are continuous, smooth, and have a finite number of discontinuities. They may also have difficulties with functions that are not integrable or have an infinite number of discontinuities.

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