Isolated Singularities and Series Expansions
Part 4: Residues
If
the function f has an isolated singularity at z0, then
it has a Laurent series expansion in some punctured disk centered at z0:
The coefficient c-1 in this expansion is called the residue of f at z0.
- Find each of the isolated singularities of the following function. (There are six.)
- Classify each of the isolated singularities of f as a removable singularity, a pole, or an essential singularity.
- Find the residue of f
at each isolated singularity. (If the residue has a complicated symbolic form,
find a numerical approximation.)
The residue command in your computer
algebra system probably will not find the residue at an essential singularity.
In these cases, you have to do it yourself.
- What is the Laurent expansion
for
g(z) = sin(2/z)
about z = 0?
- What is the residue of sin(2/z)
at z = 0?
