Math 411: Honours Complex Variables - University of Alberta

56 CHAPTER 8. THE SINGULARITIES OF A HOLOMORPHIC FUNCTION
Case 2: ˜g(z0) = 0. For z ∈ Bǫ0(z0), we have
�
�
|f(z)|= �
1
�˜g(z)
+w0
�
�
�
1
� ≥
|˜g(z)| −|w0| → ∞.
z→z0
Hence, z0 is a pole of f, again contradicting the fact that z0 is an essential singularity.
Problem 8.1.
Let D ⊂ C be open, let f : D → C be holomorphic, and let z0 ∈ D. Show that the
following are equivalent for n ∈ N:
(i) f (k) (z0) = 0 for k = 0,...,n−1 and f (n) (z0) �= 0;
(ii) there exists a holomorphic function g: D → C with g(z0) �= 0 such that f(z) =
(z −z0) n g(z) for z ∈ D.
If either condition holds, we say that z0 is a zero of f of order n.
Problem 8.2.
Let D ⊂ C be open, let f,g: D → C be holomorphic, and let z0 ∈ D be a zero of
order n for f and of order m ≥ 1 for g. Show the singularity z0 of f
g is
(i) removable if m ≤ n and
(ii) a pole of order m−n otherwise.
Problem 8.3.
Let D ⊂ C be open, let f: D → C be holomorphic, and let z0 ∈ C\D be an isolated
singularity of f.
(a) Show that, if z0 is a pole of order k of f, then it is a pole of order k+1 of f ′ .
(b) Show that exp◦f has either a removable or an essential singularity at z0.