Problem with this estimation lemma example

In summary, the conversation discusses the problem of showing that the given limit equals 0, with given conditions for R and k, and a curve C defined in terms of x and U. The speaker mentions using the fact that |e^{ikz}|=e^{-kU}, but got stuck after that. The other person suggests calculating the residues at the poles and using the residue theorem, and also suggests using the fact that |\frac{ze^{ikx}}{z^2+a^2}|\leq e^{-kU}|\frac{z}{z^2+a^2}|. The conversation ends with the speaker still unsure of what to do next.
  • #1
Jenny short
3
0
I have been trying to show that

$$\lim_{U\rightarrow\infty}\int_C \frac{ze^{ikz}}{z^2+a^2}dz = 0 $$

Where $$R>2a$$ and $$k>0$$ And C is the curve, defined by $$C = {x+iU | -R\le x\le R}$$

I have tried by using the fact that

$$|\int_C \frac{ze^{ikz}}{z^2+a^2}dz| \le\int_C |\frac{ze^{ikz}}{z^2+a^2}|
|dz|$$

I want to use the fact $$|e^{ikz}|=e^{-kU}$$

However I got really stuck after that. I would really appreciate help
 
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  • #2
I do not understand your description of the curve C. Anyhow, [itex]z^{2}+a^{2}=(z-ia)(z+ia) [/itex], so you have poles in ia and -ia. The residue at ia is [itex]\frac{iae^{-ka}}{2ia}=\frac{e^{-ka}}{2} [/itex]. Now you just have to calculate the other residue and use the residue theorem...
 
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  • #3
[itex]|\frac{ze^{ikx}}{z^2+a^2}|\leq e^{-kU}|\frac{z}{z^2+a^2}|[/itex]

Does that help?
 
  • #4
mathman said:
[itex]|\frac{ze^{ikx}}{z^2+a^2}|\leq e^{-kU}|\frac{z}{z^2+a^2}|[/itex]

Does that help?
I've done that, but I'm suck on what to do after that
 
  • #5
To continue what I said above: As long as C is given by [itex] \vert z\vert=R[/itex] with [itex]R>\vert a \vert [/itex], the value of the integral is given by [itex]2\pi i\sum Res_{\vert z \vert <R} [/itex].
 
Last edited:
  • #6
More information: If neither ia or -ia is inside C, then the function is analytic there, thus the integral must be 0.
 

Related to Problem with this estimation lemma example

1. What is an estimation lemma?

An estimation lemma is a mathematical tool used in the field of approximation algorithms to show that a problem can be solved efficiently by providing an upper bound or a lower bound on the problem's solution.

2. How does the estimation lemma work?

The estimation lemma works by breaking down a complex problem into smaller, more manageable subproblems and using mathematical techniques to estimate the solution for each subproblem. These estimates are then combined to provide an overall estimation for the original problem.

3. What are some examples of problems that can be solved using the estimation lemma?

The estimation lemma can be applied to a wide range of problems in various fields, such as computer science, engineering, and statistics. Some examples include the traveling salesman problem, the knapsack problem, and the shortest path problem.

4. What are the benefits of using the estimation lemma?

The estimation lemma allows for the efficient and accurate approximation of solutions for complex problems. It also helps in analyzing the time and space complexity of algorithms, making it a valuable tool in the development of efficient algorithms.

5. Are there any limitations to using the estimation lemma?

While the estimation lemma is a powerful tool, it is not applicable to all problems. In some cases, the estimation may not be precise enough, and in other cases, it may be computationally expensive to calculate the estimates. Additionally, the estimation lemma may not provide the optimal solution for a problem.

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