Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
For the converse, let g: Br(z0) → C be holomorphic such that g(z0) �= 0 and<br />
f(z) = g(z)<br />
(z−z0) k for z ∈ Br(z0)\{z0}. Let g(z) = �∞ n=0bn(z−z0) n for z ∈ Br(z0)<br />
be the Taylor series <strong>of</strong> g, so that<br />
f(z) =<br />
∞�<br />
bn(z −z0) n−k<br />
n=0<br />
for z ∈ Br(z0)\{z0}. The uniqueness <strong>of</strong> the Laurent representation yields that<br />
an = bn+k for n ≥ −k and an = 0 for n < −k.<br />
(iii) This follows from (i) and (ii) by elimination.<br />
Examples.<br />
1. Let<br />
Then f has the Laurent representation<br />
1<br />
−<br />
f: C\{0} → C, z ↦→ e z2 .<br />
f(z) =<br />
∞� (−1) n 1<br />
n! z2n n=0<br />
for z ∈ C\{0} and thus has an essential singularity at 0.<br />
2. Let<br />
so that<br />
f: C\{0} → C, z ↦→ ez −1<br />
z3 ,<br />
f(z) = 1<br />
z 3<br />
∞�<br />
n=1<br />
z n<br />
n! =<br />
∞� zn−3 n=1<br />
for z ∈ C\{0}. Hence, f has a pole <strong>of</strong> order two at 0.<br />
Remark. TheLaurentrepresentation<strong>of</strong>aholomorphicfunctiononanannulusAr,R(z0)<br />
depends not only on z0, but also on r and R.<br />
Example. Consider the function<br />
f: C\{1,3} → C, z ↦→<br />
and note that<br />
f(z) = 1 1<br />
−<br />
1−z 3−z .<br />
Then f has the following Laurent representations:<br />
n!<br />
2<br />
z 2 −4z +3 ,<br />
63