Complex Analysis: Theory and Applications · Problem 7.3.18

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · z-transform-laurent-series

Problem

(The

Z

-transform): Let

\{a_n\}

be a sequence of complex numbers satisfying the growth condition

|a_n| \leq MR^n

for

n = 0, 1, \ldots

and for some fixed positive values

M

and

R

. Then the

Z

-transform of the sequence

\{a_n\}

is the function

F(z)

defined by

\mathcal{Z}(\{a_n\}) = F(z) = \displaystyle\sum_{n=0}^{\infty} a_n z^{-n}

. ((a)) Prove that

F(z)

converges for

|z| > R

. ((b)) Find

\mathcal{Z}(\{a_n\})

for (i.)

a_n = 2

. (ii.)

a_n = \dfrac{1}{n!}

. (iii.)

a_n = \dfrac{1}{n+1}

. (iv.)

a_n = 1

, when

n

is even, and

a_n = 0

when

n

is odd. ((c)) Prove that

\mathcal{Z}(\{a_{n+1}\}) = z\!\left[\mathcal{Z}(\{a_n\}) - a_0\right]

. This relation is known as the shifting property for the

Z

-transform.

Intuitive explanation

The $Z$ -transform converts a discrete sequence into a power series in $z^{-1}$ , much like how the Laplace transform handles continuous functions. The key insight is recognizing this as a geometric series (or related series) in disguise: the growth bound on $|a_n|$ ensures convergence for large enough $|z|$ , and specific sequences often sum to familiar closed forms. Part (c) reveals a crucial algebraic property—shifting indices in the sequence corresponds to a simple algebraic manipulation of the transform, making it a powerful tool for solving recurrence relations.

Intuitive explanation

Related theory

Laurent Series And Isolated Singularities

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