8.1 Compositions with Sequences

8.1   Definition (Composition) Let $a\colon n\mapsto a(n)$ be a complex sequence. Let $g\colon S\to \mbox{{\bf C}}$ be a function such that $\mbox{{\rm dom}}(g)=S\subset\mbox{{\bf C}}$, and $a(n)\in S$ for all $n\in\mbox{{\bf N}}$. Then the composition $g\circ a$ is the sequence such that $(g\circ a)(n)=g\left(a(n)\right)$ for all $n\in\mbox{{\bf N}}$. If $a$ is a sequence, I will often write $a_n$ instead of $a(n)$. Then

$$a=\{a_n\}\mbox{$\hspace{1ex}\Longrightarrow\hspace{1ex}$}g\circ a=\{g(a_n)\}.$$

8.2   Examples. If

$$f = \left\{ {1\over 2^n} \right\} =\left\{ 1,{1\over 2}, {1\over 4},\cdots \right\}$$

and $${ g(z) = {1\over 1+z}}$$ for all $z \in \mbox{{\bf C}}\setminus \{-1\}$, then

$$g\circ f = \left\{ {1\over 1+{1\over 2^n}}\right\} = \left\{ ... ...ht\} =\left\{ {1\over 2},{2\over 3},{4\over 5},\cdots\right\}.$$

Figure a) below shows representations of $x, v\circ x$ and $K\circ x$ where

$$x(n) = x_n={2\over 5}+{4\over 5}i+\left( {{4-21i}\over {25}}\right)^{n+1} \mbox{ for all } n\in\mbox{{\bf N}},$$

$$v(z) = {1\over z} \mbox{ for all }z\in\mbox{{\bf C}}\backslash\{0\},$$

$$K(z) = z+{z\over {\vert z\vert}}\mbox{ for all }z\in\mbox{{\bf C}}\backslash\{0\}.$$ (8.3)

I leave it to you to check that $${ \{x_n\}\to {2\over 5}+{4\over 5}i}$$, and $${\{v(x_n)\}\to v\left({2\over 5}+{4\over 5}i\right)}$$, and $${\{K(x_n)\}\to K\left({2\over 5}+{4\over 5}i\right)}$$. Figure b) shows representations for $a$ and $K\circ a$ where

$$a(n)=a_n=\left({{7-23i}\over {25}}\right)^n,$$

and $K$ is defined as in (8.3). Here it is easy to check that $\{a_n\}\to 0$. From the figure, $\{K(a_n)\}$ doesn't appear to converge.

8.4   Exercise. A Let $\alpha$ be a non-zero complex number with $0<\vert\alpha\vert<1$. Let

$$f_\alpha (n)=\alpha^n+{{\alpha^n}\over {\vert\alpha^n\vert}} \mbox{ for all } n\in\mbox{{\bf N}}.$$

Under what conditions on $\alpha$ does $f_\alpha$ converge? What does it converge to? (Your answer should show that the sequence $\{K(a_n)\}$ 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 $\mbox{{\bf C}}$, and whose codomain is $\mbox{{\bf C}}$. I will consider functions from $\mbox{{\bf R}}$ to $\mbox{{\bf R}}$ to be complex functions by identifying a function $f\colon S\to\mbox{{\bf R}}$ with a function $f\colon S\to\mbox{{\bf C}}$ in the expected manner.