Meromorphic Functions

definition (Meromorphic):

A function f:ΩC 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)=(n1k=0f(k)(z0)k!(zz0)k)+Rn(z)(zz0)n, where Rn is analytic.