How to Determine the Order of the Pole

To determine the order of a pole of a function, you need to follow a few steps.

First, identify the point at which you suspect the pole exists, usually denoted as a. A pole is a specific type of singularity where the function approaches infinity.

Next, you can analyze the behavior of the function as it approaches this point. This is often done by examining the function in a vicinity of the point a.

To formally find the order:

1. Express the function in the form of a fraction: f(z) = g(z) / h(z), where g(z) and h(z) are analytic functions near a.
2. Find the zeros of h(z) near the point a and determine the multiplicity of the zero. The number of times h(z) goes to zero at a indicates the order of the pole.
3. If h(z) has a zero of order n at a, then a is a pole of order n of the function f(z).

For example, consider the function f(z) = 1/(z - 1)^3. The function has a pole at z = 1, and since (z - 1) has a zero of order 3, we can conclude that z = 1 is a pole of order 3.

Thus, by following these steps, you can systematically determine the order of a pole for any rational function.