276 CHAPTER 7 • TAYLOR AND LAURENT SERIES
- The Z-transform. Let {an} be a sequence of complex numbers satisfying the growth
condition lanl :::; M Rn for n = 0, 1, ... and for some fixed positive values M and
R. T hen t he Z-transform of the sequence {an} is the function F(z} defined by
00
Z({an}) = F (z) = L: anz-n.
n=O
(a) Prove t hat F (z) converges for lzl > R.
(b) Find Z({a .. }) for
i. a,, = 2.
ii. an= ;h.
••• 1
UJ. an= n+l ·
iv. a,. = 1, when n is even , and an = 0 when n is odd.
(c} Prove that Z ({an+i}) = z (Z ({an}) - ao]. This relation is known as
the shifting property for the Z-transform.
00
19. Use the Weierstrass M -test to show that the series 2:: C- n (z-a)-n of T heorem
n=l- 7 converges uniformly on the set { z : lz -al ~ s } as claimed.
- Verify the following claims made in this section.
(a) T he series in Equation (7-27) converges uniformly for z E C2.
(b) The validity of Equation (7-28) , according to Corollary 4.2.
(c) The series in Equation (7-28) converges uniformly for z E C1.7.4 Singularities, Zeros, and Poles
Recall that the point a is called a singular point, or singularity, of the com-
plex function f if f is not analytic at the point a, but every neighborhood DR (a)
of a contains at lea.st one point at which f is analytic. For example, the functionf (z) = 1 : , is not analytic at a = 1 but is analytic for all other values of z.
Thus, the point a = 1 is a singular point of f. As another example, consider
the function g (z) =Log z. We showed in Section 5.2 that g is analytic for all z
except at the origin and at the points on the negative real axis. Thus, the origin
and each point on the negative real axis are singularities of g.
The point a is called an isolated singularity of a complex function f if
f is not analytic at a but there exists a real number R > 0 such t hat f is
analytic everywhere in the punctured disk Dn (a). The function f (z) = 1 :.
has an isolated singularity at a= 1. The function g (z) =Log z, however, has a
singularity at a = 0 (or at any point of the negative real axis) tha.t is not isolated,
because any neighborhood of a contains points on the negative real axis, and
g is not analytic at those points. Functions with isolated singularities have a