Orders of poles in the complex plane.

[jameson2]

Homework Statement

For a function f(z)= [e^(2*pi*i*a*z)] / [1 + z^2] I need to find the order of the poles at i and -i.
(I'm pretty sure these are the only poles.)

Homework Equations

The Attempt at a Solution

I'm not totally clear on how I go about finding the orders. I have a vague idea: I can factorise the bottom line to get (z-i)(z+i) and I think this means that both poles are of order 1.
But I want to try do the question by using the power series of the numerator and denominator, by comparing the number of "zero coefficients" in each power series.
Anyone know how to do this method?

 

[mighty2000]

To find the order of a pole, you can use the formula:
order = n, where n is the highest power of z in the denominator of the Laurent series expansion of the function at that pole. In this case, the denominator is (1+z^2), so the highest power of z is 2. Therefore, the poles at i and -i are both of order 2.

To use the power series method, you can expand the numerator and denominator separately and compare the number of "zero coefficients." However, this method may not always be reliable as it depends on the specific function and the choice of power series. It is better to use the formula mentioned above to determine the order of poles.

Hope this helps. Best of luck with your problem!