ECE 3250
THE
z
TRANSFORM
Fall 2008
1. Power series
We’ll have occasion to consider infinite series of the form
∞
X
n
=
∞
c
n
z

n
,
where
z
is a complex variable and the
c
n
are given complex numbers. We’ll want to know
whether the series converges for at least some
z
∈
C
and, if so, for exactly what values
of
z
the series converges.
The special form of power series makes for a tidy theory of
convergence.
Fact 1:
If
R
o
>
0 is such that the sequence
{
c
n
R

n
o
:
n
≥
0
}
is bounded, then the
series
∞
X
n
=0
c
n
z

n
converges for every
z
∈
C
satisfying

z

> R
o
. Similarly, if
R
o
>
0 is such that the sequence
{
c
n
R

n
o
:
n <
0
}
is bounded, then the series

1
X
n
=
∞
c
n
z

n
converges for every
z
∈
C
satisfying

z

< R
o
.
Proof:
Consider the first assertion. Suppose
R
o
>
0 is such that the sequence
{
c
n
R

n
o
:
n
≥
0
}
is bounded from above in magnitude, say by
M >
0. If

z

> R
o
, then
∞
X
n
=0

c
n
z

n

=
∞
X
n
=0

c
n
R

n
o

(
R
o
/

z

)
n
≤
M
∞
X
n
=0

(
R
o
/

z

)
n
=
M/
(1

R
o
/

z

)
,
where the last equality holds by geometricseries reasoning since
R
o
/

z

<
1.
It follows
that the infinite sequence
{
c
n
z

n
}
is absolutely summable and is therefore summable by
Facts 3 and 7 from the handout on sequences and series. The proof of the second assertion
is similar.
Fact 2:
There exists some
z
∈
C
for which the series
∞
X
n
=0
c
n
z

n
converges if and only if there exists some
R
o
>
0 for which the sequence
{
c
n
R

n
o
:
n
≥
0
}
is bounded. Similarly, there exists some
z
∈
C
for which the series

1
X
n
=
∞
c
n
z

n
converges if and only if there exists some
R
o
>
0 for which the sequence
{
c
n
R

n
o
:
n <
0
}
is bounded.
Proof:
The “if” part is just Fact 1. As for the “only if,” suppose
z
o
∈
C
is such that
P
∞
n
=0
c
n
z

n
o
(respectively,
P

1
n
=
∞
c
n
z

n
) converges.
Then the sequence
{
c
n

z
o


n
:
n
≥
0
}
(respectively,
{
c
n

z
o


n
:
n <
0
}
) must be bounded, so taking
R
o
=

z
o

proves
the assertion(s).
1
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2
Suppose now that
{
c
n
:
n
∈
Z
}
are given complex numbers, and suppose there exists
R
o
>
0 such that the sequence
{
c
n
R

n
o
}
is bounded. Define
R
a
as follows:
R
a
= inf
{
R
o
>
0 :
{
c
n
R

n
o
:
n
≥
0
}
is bounded
}
.
Observe that
R
a
= 0 is possible. Similarly, define
R
b
via
R
b
= sup
{
R
o
>
0 :
{
c
n
R

n
o
:
n <
0
}
is bounded
}
.
Observe that
R
b
=
∞
is possible. Taken together, Facts 1 and 2 reveal that the series
∞
X
n
=0
c
n
z

n
converges for every
z
∈
C
satisfying

z

> R
a
and diverges for every
z
satisfying

z

< R
a
.
To see why it diverges for such
z
, note that if the series converged for some
z
o
with

z
o

=
R
o
< R
a
, then
{
c
n
R

n
o
:
n
≥
0
}
would be bounded, and Fact 1 along with the
definition of
R
a
would imply that
R
o
≥
R
a
, a contradiction. Similarly, the series

1
X
n
=
∞
c
n
z

n
converges if

z

< R
b
and diverges if

z

> R
b
.
The infinite series
P
∞
n
=
∞
c
n
z

n
therefore converges at least for
z
values in the region
R
a
<

z

< R
b
. Such
z
values exist if and only if
R
a
< R
b
, in which case the indicated
set of
z
values constitutes an
annular region
centered on
z
= 0 in the complex plane. The
series may or may not converge for some
z
values satisfying

z

=
R
a
or

z

=
R
b
, but we
pay little attention to those borderline values. One can show that the series diverges for
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