8.1
Definition (Composition)
Let
![$a\colon n\mapsto a(n)$](img584.gif)
be a complex sequence. Let
![$g\colon S\to \mbox{{\bf C}}$](img585.gif)
be a
function such that
![$\mbox{{\rm dom}}(g)=S\subset\mbox{{\bf C}}$](img586.gif)
, and
![$a(n)\in S$](img587.gif)
for all
![$n\in\mbox{{\bf N}}$](img9.gif)
. Then the
composition ![$g\circ a$](img588.gif)
is
the sequence such that
![$(g\circ
a)(n)=g\left(a(n)\right)$](img589.gif)
for all
![$n\in\mbox{{\bf N}}$](img9.gif)
. If
![$a$](img590.gif)
is a sequence, I will often
write
![$a_n$](img526.gif)
instead of
![$a(n)$](img591.gif)
. Then
8.2
Examples.
If
and
![$\displaystyle { g(z) = {1\over 1+z}}$](img594.gif)
for all
![$z \in \mbox{{\bf C}}\setminus \{-1\}$](img595.gif)
,
then
Figure a) below shows representations of
and
where
I leave it to you to check that
![$\displaystyle { \{x_n\}\to {2\over 5}+{4\over 5}i}$](img605.gif)
, and
![$\displaystyle {\{v(x_n)\}\to v\left({2\over 5}+{4\over 5}i\right)}$](img606.gif)
, and
![$\displaystyle {\{K(x_n)\}\to K\left({2\over 5}+{4\over 5}i\right)}$](img607.gif)
.
Figure b) shows representations for
![$a$](img590.gif)
and
![$K\circ a$](img608.gif)
where
and
![$K$](img532.gif)
is defined as in (
8.3). Here it is easy to check that
![$\{a_n\}\to
0$](img610.gif)
. From the figure,
![$\{K(a_n)\}$](img611.gif)
doesn't appear to converge.
8.4
Exercise.
A
Let
![$\alpha$](img12.gif)
be a
non-zero complex number with
![$0<\vert\alpha\vert<1$](img355.gif)
.
Let
Under what conditions on
![$\alpha$](img12.gif)
does
![$f_\alpha$](img615.gif)
converge? What does it converge
to? (Your answer should show that the sequence
![$\{K(a_n)\}$](img611.gif)
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
![$\mbox{{\bf C}}$](img2.gif)
,
and whose codomain is
![$\mbox{{\bf C}}$](img2.gif)
. I will consider functions from
![$\mbox{{\bf R}}$](img4.gif)
to
![$\mbox{{\bf R}}$](img4.gif)
to be
complex functions by identifying a function
![$f\colon S\to\mbox{{\bf R}}$](img616.gif)
with a function
![$f\colon S\to\mbox{{\bf C}}$](img617.gif)
in the expected manner.