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Pyroadept

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## 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