Complex analysis question: Calculus of residues

[Pyroadept]

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.

Homework Equations

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


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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

 

[mighty2000]

!


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.