Completing a Fourier Transform Integral

In summary, contour integration is a technique used to solve integrals by choosing a contour on which the integrand is complex analytic. This allows for the computation of the integral around the contour, which is related to the poles of the function within the contour. This method is often used to evaluate desired integrals by choosing a contour that makes the other segments either go to zero or be multiples of the desired integral. It is recommended to pick up a book on complex analysis for a better understanding of this technique.
  • #1
ajw124
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I was wondering if anyone could help me with this integral. I've heard of contour integration but I'm unsure of how it would be used for this integral.
fourierInt.jpg
 
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  • #2
Well, I suggest you pick up a book on complex analysis. It's not something which lends itself to being taught via a single forum post. This one is nice enough and affordable.

The general idea is you choose a contour on which the integrand is complex analytic, which includes the desired integral as a segment of the contour. One then computes the integral around the contour. The value of the integral around the contour is related to the poles of the function that are within the contour. Often the actual integral which you want to evaluate is not the contour integral itself, but rather a segment integral of that contour. The contour is often chosen such that the integral along the other segments besides the ones desired either go to zero in some limit, or are multiples of the desired integral.
 
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Related to Completing a Fourier Transform Integral

1. What is a Fourier Transform Integral?

A Fourier Transform Integral is a mathematical operation that converts a function from its original domain (usually time or space) to a representation in the frequency domain.

2. How is a Fourier Transform Integral completed?

A Fourier Transform Integral is completed by performing an integration over the function with respect to the variable of interest, typically time or space.

3. What is the purpose of completing a Fourier Transform Integral?

The purpose of completing a Fourier Transform Integral is to analyze the frequency components of a given function and to understand how the function's behavior changes over different frequencies.

4. What types of functions can be transformed using a Fourier Transform Integral?

A Fourier Transform Integral can be applied to any function that is continuous and has finite energy, including waveforms, signals, and images.

5. Are there any limitations to completing a Fourier Transform Integral?

One limitation is that the function must be well-behaved, meaning that it must have a finite number of discontinuities and singularities. Additionally, the function must have finite energy in order for the integral to converge.

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