598 Chapter^8
with r' ~ lz -al ::::; R'. For any given f > 0, there exists a positive integer
N such that
for n ~ N and all z satisfying lz -al ::::; R', and also a positive integer
N' such that
for n' ~ N' and all z satisfying lz -al ~ r'. Hence we have
for n, n' ~ max(N,N') and all z such that r'::::; lz -al::::; R'. This proves
the uniform copvergence of Sn(z) + Tn1(z) to fi(z) + h(z) on the closed
ring r' :::; lz - al :::=; R'.
Remarks From the preceding discussion it follows that f(z) admits a
decomposition of the form f(z) = fi(z) + h(z), where fi(z) is analytic
for lz -al < Rand h(z) is analytic for lz -al > r. Also, it follows that
the largest region of convergence of the Laurent series can be obtained by
increasing R and decreasing r as long as f remains defined and analytic
in the enlarged ring. Therefore, excepting the trivial case of a constant
function in C, the largest possible ring will contain at least one singularity
of f on either boundary (or on both).
It is clear that if f is analytic on lz -al ::::; r, or at least locally bounded
at z =a, then A-n = 0 for every n ~ 1, and the Laurent series reduced to a
Taylor series. However, if f fails to be analytic at a (with a not removable),
yet analytic in 0 < lz -al ::::; r, then for some n, A_n #-0. In this case the
radius r can be taken as small as we please, and the Laurent expansion is
valid for 0 < lz -al < R. On the other hand, if R can be taken as large
as we please then the expansion holds for lz -al > r.
The Laurent series expansion proves to be an useful tool for the study
of isolated singularities of analytic functions, as we shall see in Chapter 9.
In many cases formula (8.18-2) does not provide the best method for
the computation of the coefficients Am, and it is of interest to prove the
uniqueness of the expansion, since this will guarantee that the coefficients
are independent of any alternative method used to obtain the Laurent
series.
Theorem 8.44 The coefficients in the Laurent expansion of an analytic
function in r < lz -al < R are uniquely determined.