** 8.1**
**Definition (Composition)**
Let

be a complex sequence. Let

be a
function such that

, and

for all

. Then the

*composition* is
the sequence such that

for all

. If

is a sequence, I will often
write

instead of

. Then

** 8.2**
**Examples.**
If

and

for all

,
then

Figure a) below shows representations of and where

I leave it to you to check that

, and

, and

.
Figure b) shows representations for

and

where

and

is defined as in (

8.3). Here it is easy to check that

. From the figure,

doesn't appear to converge.

** 8.4**
**Exercise.**
A
Let

be a
non-zero complex number with

.
Let

Under what conditions on

does

converge? What does it converge
to? (Your answer should show that the sequence

from the previous
example does not converge.)

** 8.5**
**Definition (Complex function.)**
By a

*complex function* I will mean a function whose domain is a subset of

,
and whose codomain is

. I will consider functions from

to

to be
complex functions by identifying a function

with a function

in the expected manner.