(^218) Singularities of Complex Functions
Then
_m = f /(OK-)-
1
(447)and it remains to integrate this equality.
Step 5. We combine the results of the above steps to get both (4.4.2)
and (4.4.3). Note that any simple, closed, piece-wise smooth curve can be
used instead of Cp in (4.4.3), as long as the curve is completely inside G,
the closed disk {z : \z — ZQ\ < R\\ is inside the domain enclosed by the
curve, and the point z is outside that domain.
Step 6. To prove the uniqueness, we multiply the equality
'E'kL-occi*(z-zo)k = T,'kL-ooak(z-zo)k by (z-zo)-™-^1 for some integer
m and integrate both sums term-by-term over the circle Cp. By (4.2.6), page
195, all integrals become zero except for those corresponding to k = m, and
we get cm = am. •EXERCISE 4.4.1. A (a) Verify (4-4-4)• Hint: connect the circles Cr and Cp
•with a line segment that does not pass through the point z and write the Integral
Theorem of Cauchy in the resulting simply connected domain. Then simplify the
result, keeping in mind that you integrate along the line segment twice, but in
opposite directions, and that the orientation of Cp is clockwise. (b) Justify
the term-by-term integration in (4-4-V- Hint: note that, for £ e Cp, we have
|(C — zo)/(z — zo)\ — p/\z — zo\ < 1. Then use the same argument as in the
proof of Theorem 4-3-4, Page 210. (c) Fill in the details in the proof of the
uniqueness of the expansion. Hint: once again, the key step is justifying the
term-by-term integration, and once again, you use the same arguments as in the
previous similar cases.Note that if, in the above theorem, the function is analytic for all z
satisfying \z — ZQ\ = Ri, then we can decrease R\. Similarly, we can increase
R if / is analytic for all z satisfying \z — ZQ\ = R (make sure you understand
this). In other words, with no loss of generality, we will always assume
that the annulus Ri < \z — zo\ < R is maximal, that is, each of the sets
|z — 2o| = Ri and \z — z§\ = R (assuming R < oo) contains at least one
point where the function / is not analytic. There are several types of such
points.Definition 4.4 A point ZQ € C is called an isolated singular point
or an isolated singularity of a function / if the function / is not ana-