Complex analysis question: Calculus of residues

In summary, we have a function f(z) = exp(2πiEz) / (1 + z^2) with poles at z = i and z = -i, both with pole orders of 1. The residues at these poles can be calculated using the formula a_-1 = lim (z - a).f(z), and the same formula can be used to calculate the residue at infinity by substituting z = 1/w and taking the limit as w goes to 0. The Cauchy Residue Theorem can also be used to calculate the residues.

Homework Statement

Let f(z) = exp(2πiEz) / (1 + z^2), where E is some real number.

Find the poles, their orders and the residues at each pole.

The Attempt at a Solution

Hi everyone, here's what I've done so far:

1 + z^2 = (1 + i)(1 - i)

Thus f has poles at z = i, z = -i, as f(z) is non-holomorphic at these points.

Both are poles of order 1, as lim (z - a).f(z) are defined, for a = i, -i (limit as z goes to a)

Now calculate the resdiues, use the formula:

a_-1 = lim (z - a).f(z) [limit as z goes to a]

and then Res f(z) = 2πi.a_-1

---

Is this correct? Is it acceptable to just sub in the values into the equation like this, or are you expected to work out the Laurent series?

Also, is there also a pole at z = infinity?
Can it too be subbed into the equations above?

Thanks for any help

!

Hi there, your approach is correct. You have correctly identified the poles and their orders. To calculate the residues, you can use the formula you mentioned, or you can also use the Cauchy Residue Theorem. Both methods should give you the same result.

As for the pole at infinity, yes, it can also be subbed into the equations above. In this case, you can use the substitution z = 1/w and then take the limit as w goes to 0. This will give you the residue at infinity, which is equal to the coefficient of 1/w in the Laurent series expansion around infinity.

Hope this helps! Let me know if you have any other questions.

1) What is the purpose of calculating residues in complex analysis?

The residues in complex analysis are used to evaluate complex integrals that cannot be solved using traditional methods. They provide a way to analyze the behavior of a function around its singularities, which are points where the function is not defined or behaves differently than in surrounding points.

2) How do you calculate the residue of a function at a pole?

In order to calculate the residue of a function at a pole, one must first identify the pole by finding the roots of the denominator of the function. Then, the residue can be found using the formula Res(f,c) = lim(z→c) [(z-c)f(z)], where c is the pole and f(z) is the function.

3) Can residues be used to find the values of complex integrals?

Yes, residues can be used to evaluate complex integrals. This is done by using the Residue Theorem, which states that the value of a complex integral around a closed curve is equal to 2πi times the sum of the residues of the function inside the curve.

4) How is the Cauchy Residue Theorem used in complex analysis?

The Cauchy Residue Theorem is a powerful tool in complex analysis that allows for the evaluation of complex integrals using residues. It states that if a function is analytic inside and on a simple closed curve except for a finite number of poles, then the integral of that function around the curve is equal to 2πi times the sum of the residues of the function at its poles within the curve.

5) Are there any applications of calculating residues in real-world problems?

Yes, calculating residues has various applications in real-world problems, particularly in physics and engineering. For example, it can be used to analyze the behavior of electronic circuits, to solve differential equations, and to calculate the probability of certain events in quantum mechanics. It is also used in the field of signal processing to analyze signals and filters.

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