592 Chapter 8
1. If r == 0, the series (8.15-2) converges absolutely for every finite value
of(. Hence the series (8.15-1) converges absolutely for every z in C*
except z = a.2. If 0 < r < oo, the series (8.15-2) converges absolutely for every ( such
that ICI < 1/r and diverges for ICI > 1/r. Hence the series (8.15-1)
converges absolutely for lz -al >rand diverges for lz -al < r. Also,
(8.15-2) converges uniformly for ICI ::;: 1/r' < 1/r, so that (8.15-1)
converges uniformly for lz -al 2':: r^1 > r.
- If r = oo, the series (8.15-2) converges only for ( = 0. Hence (8.15-1)
converges only for z = oo; i.e., it diverges for all finite values of z.
Corollary 8.20 For 0 ::;: r < oo the series (8.15-1) converges uniformly
on every compact subset of the region of convergence.
Corollary 8.21 For 0::;: r < oo, let J(z) = z:::=o cn(z-a)-n. Then J(z)
is analytic at every finite point of the region lz -al > r.
8.16 Laurent Series
Definition 8.4 Series of the form
+oo oo oo
L Am(z - ar = L A-n(z - a)-n + L An(z -at
m=-oo n=l n=O(8.16-1)where a is a constant and {An}, {A-n} are sequences of constants, are
called Laurent _series after P. A. Laurent (1813-1854 ), who introduced those
series in 1843 [21 ]. t As the notation suggests, they are interpreted as the
sum of a series of negative integral powers of z - a and of an ordinary series
in nonnegative integral powers of z - a. \
A Laurent series is convergent for a value of z iff each of the series oh
the right-hand side of (8.16-1) converges for that value of z. It is called
divergent if at least one of the series in (8.16-1) diverges for that value of
z. Hence the notation I:~~-oo Am(z - a)m has the following meaning:
+oo h k
"· L..J Am(z-ar= h-+oo lim "A-n(z-a)-n+ L..J k-+oo lim L..J "An(z-at
m=-oo n=l n=OfHowever, Weierstrass knew of this type of series in 1841 but did not publish
his results.