Proving Improper Integral with Complex Analysis

In summary, to prove an improper integral using complex analysis, the function is first extended to the complex plane and then evaluated using techniques such as Cauchy's Integral Theorem or Residue Theorem. An improper integral is an integral with one or both limits of integration being infinite or with a singularity at one of the limits. Complex analysis is useful for proving improper integrals because it allows for the extension of the integrand to the complex plane, leading to simpler and more elegant solutions. Common techniques in complex analysis for proving improper integrals include Cauchy's Integral Theorem and Residue Theorem. However, complex analysis may not be applicable to all types of improper integrals, and other techniques, such as real analysis methods,
  • #1
jusy1
2
0
Hi everybody

I was trying to prove that [tex]\int_{-\infty}^{\infty}e^{\imath (k - k') x}dx = 2\pi\delta(k-k')[/tex] by solving [tex]\lim_{L\rightarrow \infty} \int_{-L}^{L}e^{\imath (k - k') x}dx[/tex]

knowing that [tex]\delta(x)=\lim_{g\rightarrow \infty}\frac{\sin(gx)}{\pi x}[/tex]

But is there a way of proving this result using complex analysis?
 
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  • #2
Wouldn't it be easier to prove this considering piecewise, i.e. consider k≠k' and then consider k=k'?
 

1. How do you use complex analysis to prove an improper integral?

To use complex analysis to prove an improper integral, we first extend the function to the complex plane and then use techniques such as Cauchy's Integral Theorem or Residue Theorem to evaluate the integral. These theorems allow us to express the integral as a contour integral, which can then be evaluated using complex analysis techniques.

2. What is an improper integral?

An improper integral is an integral where one or both of the limits of integration are infinite, or when the integrand has a singularity at one of the limits. These integrals are not defined in the traditional sense, so techniques such as complex analysis are often used to evaluate them.

3. Why is complex analysis useful for proving improper integrals?

Complex analysis is useful for proving improper integrals because it allows for the extension of the integrand to the complex plane, which can then be used to calculate the integral using techniques such as contour integration. This can often lead to simpler and more elegant solutions compared to traditional real analysis methods.

4. What are some common techniques in complex analysis for proving improper integrals?

Some common techniques in complex analysis for proving improper integrals include the Cauchy's Integral Theorem, which states that the integral of a function along a closed contour is equal to the sum of the residues of the function at its singularities inside the contour. The Residue Theorem is also commonly used, which states that the integral of a function around a closed contour is equal to 2πi times the sum of the residues of the function at its poles inside the contour.

5. Can complex analysis be used for all types of improper integrals?

No, complex analysis may not always be applicable to all types of improper integrals. It is best suited for integrals where the integrand can be extended to the complex plane, and where techniques such as Cauchy's Integral Theorem or Residue Theorem can be used. Other techniques, such as real analysis methods, may be better suited for other types of improper integrals.

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