definition (Meromorphic):
A function f:Ω→C is meromorphic iff there exists a sequence {zn} such that
- {zn} has no limit points in Ω.
- f is holomorphic in Ω∖{zn}.
- f has poles at the points {zn}.
Equivalently, f is holomorphic on Ω with a discrete set of points delete which are all poles of f.
theorem (Meromorphic implies rational):
Meromorphic functions on C are rational functions.
proof (?):
Consider f(z)−P(z), subtracting off the principal part at each pole z0, to get a bounded entire function and apply Liouville.
theorem (Improved Taylor Remainder Theorem):
If f is analytic on a region Ω containing z0, then f can be written as f(z)=(n−1∑k=0f(k)(z0)k!(z−z0)k)+Rn(z)(z−z0)n, where Rn is analytic.