Thread: Laurent expansions

Find all Laurent expansions centered at z0 = 0 for f(z) = 1/[z^2(z^3 + 1)] and find the region of convergence for each expansion.

For this problem, I start with rewrite f(z) = 1/(z^2) *1/ [1-(-z^3)] then use geometric series to write laurent series representation for 1/ [1-(-z^3)]. I am stuck after that. Please help. Thank you.

Originally Posted by conmeo

Find all Laurent expansions centered at z0 = 0 for f(z) = 1/[z^2(z^3 + 1)] and find the region of convergence for each expansion.

For this problem, I start with rewrite f(z) = 1/(z^2) *1/ [1-(-z^3)] then use geometric series to write laurent series representation for 1/ [1-(-z^3)]. I am stuck after that. Please help. Thank you.

There are two cases:

1. If then .

2. If then you need to consider the following:

.