Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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56 CHAPTER 8. THE SINGULARITIES OF A HOLOMORPHIC FUNCTION<br />
Case 2: ˜g(z0) = 0. For z ∈ Bǫ0(z0), we have<br />
�<br />
�<br />
|f(z)|= �<br />
1<br />
�˜g(z)<br />
+w0<br />
�<br />
�<br />
�<br />
1<br />
� ≥<br />
|˜g(z)| −|w0| → ∞.<br />
z→z0<br />
Hence, z0 is a pole <strong>of</strong> f, again contradicting the fact that z0 is an essential singularity.<br />
Problem 8.1.<br />
Let D ⊂ C be open, let f : D → C be holomorphic, and let z0 ∈ D. Show that the<br />
following are equivalent for n ∈ N:<br />
(i) f (k) (z0) = 0 for k = 0,...,n−1 and f (n) (z0) �= 0;<br />
(ii) there exists a holomorphic function g: D → C with g(z0) �= 0 such that f(z) =<br />
(z −z0) n g(z) for z ∈ D.<br />
If either condition holds, we say that z0 is a zero <strong>of</strong> f <strong>of</strong> order n.<br />
Problem 8.2.<br />
Let D ⊂ C be open, let f,g: D → C be holomorphic, and let z0 ∈ D be a zero <strong>of</strong><br />
order n for f and <strong>of</strong> order m ≥ 1 for g. Show the singularity z0 <strong>of</strong> f<br />
g is<br />
(i) removable if m ≤ n and<br />
(ii) a pole <strong>of</strong> order m−n otherwise.<br />
Problem 8.3.<br />
Let D ⊂ C be open, let f: D → C be holomorphic, and let z0 ∈ C\D be an isolated<br />
singularity <strong>of</strong> f.<br />
(a) Show that, if z0 is a pole <strong>of</strong> order k <strong>of</strong> f, then it is a pole <strong>of</strong> order k+1 <strong>of</strong> f ′ .<br />
(b) Show that exp◦f has either a removable or an essential singularity at z0.