The order shouldn't be too hard and I assume you have a formula (limit?) to calculate the residue? What have you tried?
Can someone help me please
Thanks
Edgar
1. For each of the following functions, find the order of the pole that the
function has at the point a and find the residue.
(a) z−3/[(z−1)(z−2)] , a = 2
(b) exp[iz]/z^6+1 , a = exp(ipie/6)
(c) z^2/z^2+1 , a = i
(d) 1/(z−1)^3(z−2) , a = 1
(By the way, is "pi" not "pie.")
Technically what we want to do is expand each function in a Laurent series and note what largest (negative) power of (z - a) appears at. However we can simply eyeball this, as all we need to do is count how many powers of (z - a) appear in the denominator. (This is assuming the function doesn't contain a function g(z) that has poles of its own.)
The residue is the coefficient of the term in the Laurent expansion. If the pole z = a is simple (order 1) then the residue is .
a)
This expression has a pole of order 1 at z = 2.
The residue then may be calculated as
b)
Note that contains no poles.
So the only poles are from
As only appears once in the denominator, this is a pole of order 1.
So the residue will be:
I am having a problem calculating this for some reason. I'll get back to you on it once I figure out the problem.
c)
Again we see that z = i is a pole of order 1.
So the residue will be:
.
-Dan