Math 411: Honours Complex Variables - University of Alberta

88CHAPTER13. FUNCTIONTHEORETICCONSEQUENCESOFTHERESIDUETHEOREM
Definition. Let D ⊂ C be open. A meromorphic function on D is a holomorphic
function f: D\P(f) → C such that P(f) ⊂ D is discrete in D.
Remark. If D is open and connected, and f,g: D → C are holomorphic, then f
g is
meromorphic if g is not identically zero.
Proposition 13.2. Let D ⊂ C be open, and let f be meromorphic on D. Then,
for each z0 ∈ D, there exist ǫ > 0 with Bǫ(z0) ⊂ D and holomorphic functions
for z ∈ Bǫ(z0)\{z0}.
g,h: Bǫ(z0) → C such that f(z) = g(z)
h(z)
Proof. If z0 is not a pole of f, the claim is clear.
Otherwise, choose ǫ > 0 so small that Bǫ(z0) ⊂ D and P(f)∩Bǫ(z0) = {z0}. We
can then find a holomorphic function g: Bǫ(z0) → C with g(z0) �= 0 and k ∈ N such
that
f(z) = g(z)
(z −z0) k
for z ∈ Bǫ(z0)\{z0}. Setting h(z) := (z −z0) k yields the claim.
Definition. It is convenient to define the set of extended complex numbers C∞ =
C ∪ {∞} and extend the domain of meromorphic functions to include their poles,
assigning the function value ∞ at the poles. Here ω ·∞ is identified with ∞ for all
ω ∈ ∂D. Furthermore, we define 1/0 = ∞ and 1/∞ = 0.
Definition. We can reinterpret C∞ as the unit sphere S 2 in R 3 by connecting every
point x + iy in the xy plane (which we identify with C) to the north pole (0,0,1)
with a straight line that intersects S 2 at a point P(x+iy). This defines an injective
map P : C∞ → S 2 . Under this mapping, 0 maps to the south pole, ∞ maps to the
north pole, and the unit circle ∂D maps to the equator. One can readily show that P
is continuous, and that the inverse P −1 : S 2 → C∞ is also continuous; this allows us
to identify C∞ with the Riemann sphere S 2 .