The Residue Theorem To Evaluate Integrals

In summary, the conversation discusses the use of the residue theorem to solve a problem involving square roots and the evaluation of a complex integral. The speaker is unsure if they are on the right track and seeks guidance on choosing a branch and contour for the theorem. They receive advice on using a keyhole contour and are reminded to show that certain parts of the contour will have negligible contribution to the integral. The final result is provided as $\frac{\pi}{2\sqrt{2}}\sqrt{\sqrt{5}-1}$.
  • #1
joypav
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I have no idea if this is in the right direction. I know I am going to need the summation of the residues to use the theorem. I found the residues using the limit, but do I need to change these using the euler formula?

We are supposed to be working problems at home and I am getting a bit lost as the semester goes on. I would really appreciate some help or a push in the right direction!
 

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  • #2
Square roots are multi-valued, so you have to choose a branch $\log z$ first. Also, you need to choose a contour for which the residue theorem can be applied. Consider using a keyhole contour.
 
  • #3
Okay, yeah. Something more like this? Lacking details of course, but is this the idea?
View attachment 7336
Then I will have four integrals to compute, set equal to 2(pi)i*sum of the residues.
 

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  • #4
It looks right so far. Now show that the integral of $f$ becomes negligible over certain parts of the contour.
 
  • #5
Thanks!
 
  • #6
For completion, the integral evaluates to

$$\boxed{\frac{\pi}{2\sqrt{2}}\sqrt{\sqrt{5}-1}}$$
 

FAQ: The Residue Theorem To Evaluate Integrals

1. What is the residue theorem and how does it work?

The residue theorem is a powerful mathematical tool used to evaluate complex integrals that are difficult to solve using traditional methods. It states that the integral of a function around a closed curve is equal to the sum of the residues of the function at its singular points inside the curve. This allows us to simplify complex integrals into simpler ones, making them easier to solve.

2. How do I determine the residues of a function?

To determine the residues of a function, we need to find the singular points of the function, which are the points where the function is not analytic. These points can be found by setting the denominator of the function equal to zero and solving for the variable. Once we have the singular points, we can use the formula for residues to calculate their values.

3. Can the residue theorem be used for any type of integral?

The residue theorem is specifically designed for integrals with complex-valued functions. It cannot be used for real-valued integrals. Additionally, the function must have singular points inside the closed curve for the theorem to be applicable.

4. How do I know if I need to use the residue theorem to evaluate an integral?

If you are given a complex integral and are having difficulty solving it using traditional methods, it may be a good indication that the residue theorem should be used. You should also check for singular points inside the closed curve to ensure the theorem is applicable.

5. Can the residue theorem be used for multivariable integrals?

No, the residue theorem is designed for single-variable integrals only. It cannot be applied to multivariable integrals. However, there are other theorems and methods that can be used to evaluate multivariable integrals.

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