3.1 Functions  (Page 2/2)

Many functions, though not all by any means, are defined by a single equation:

(How does this last equation define a function?)

(How does this equation determine a function?)

There are various types of functions, and they can be combined in a variety of ways to produce other functions.It is necessary therefore to spend a fair amount of time at the beginning of this chapter to present these definitions.

If $f$ and $g$ are two complex-valued functions with the same domain $S,$ i.e., $f:S\to C$ and $g:S\to C,$ and if $c$ is a complex number, we define $f+g,fg,f/g$ (if $g\left(x\right)$ is never 0), and $cf$ by the familiar formulas:

$$(f+g)\left(x\right)=f\left(x\right)+g\left(x\right),$$

$$\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right),$$

$$(f/g)\left(x\right)=f\left(x\right)/g\left(x\right),$$

and

$$\left(cf\right)\left(x\right)=cf\left(x\right).$$

If $f$ and $g$ are real-valued functions, we define functions $max(f,g)$ and $min(f,g)$ by

$$[max(f,g\left)\right]\left(x\right)=max\left(f\right(x),g(x\left)\right)$$

(the maximum of the numbers $f\left(x\right)$ and $g\left(x\right)$ ), and

$$[min(f,g\left)\right]\left(x\right)=min\left(f\right(x),g(x\left)\right),$$

(the minimum of the two numbers $f\left(x\right)$ and $g\left(x\right)$ ).

If $f$ is either a real-valued or a complex-valued function on a domain $S,$ then we say that $f$ is bounded if there exists a positive number $M$ such that $\left|f\right(x\left)\right|\le M$ for all $x\in S.$

There are two special types of functions of a real or complex variable, the even functions and the odd functions.In fact, every function that is defined on all of $R$ or $C$ (or, more generally, any function whose domain $S$ equals $-S$ ) can be written uniquely as a sum of an even part and an odd part.This decomposition of a general function into simpler parts is frequently helpful.

A function $f$ whose domain $S$ equals $-S,$ is called an even function if $f(-z)=f\left(z\right)$ for all $z$ in its domain. It is called an odd function if $f(-z)=-f\left(z\right)$ for all $z$ in its domain.

We next give the definition for perhaps the most familiar kinds of functions.

A nonzero polynomial or polynomial function is a complex-valued function of a complex variable, $p:C\to C,$ that is defined by a formula of the form

$$p\left(z\right)=\sum _{k=0}^n{a}_k{z}^k=a_0+a_{1}z+a_2{z}^2+...+a_n{z}^n,$$

where the $a_k$ 's are complex numbers and $a_n\ne 0.$ The integer $n$ is called the degree of the polynomial $p$ and is denoted by $\text{deg}\left(p\right).$ The numbers $a_0,a_1,...,a_n$ are called the coefficients of the polynomial.The domain of a polynomial function is all of $C;$ i.e., $p\left(z\right)$ is defined for every complex number $z.$

For technical reasons of consistency, the identically 0 function is called the zero polynomial . All of its coefficients are 0 and its degree is defined to be $-\infty .$

A rational function is a function $r$ that is given by an equation of the form $r\left(z\right)=p\left(z\right)/q\left(z\right),$ where $q$ is a nonzero polynomial and $p$ is a (possibly zero) polynomial. The domain of a rational functionis the set $S$ of all $z\in C$ for which $q\left(z\right)\ne 0,$ i.e., for which $r\left(z\right)$ is defined.

Two other kinds of functions that are simple and important are step functions and polygonal functions.

Let $[a,b]$ be a closed bounded interval of real numbers. By a partition of $[a,b]$ we mean a finite set $P=\{x_0<x_1<...<x_n\}$ of $n+1$ points, where $x_0=a$ and $x_n=b.$

The $n$ intervals $\left\{[x_{i-1},x_i]\right\},$ for $1\le i\le n,$ are called the closed subintervals of the partition $P,$ and the $n$ intervals $\left\{(x_{i-1},x_i)\right\}$ are called the open subintervals of $P.$

We write $\parallel P\parallel$ for the maximum of the numbers (lengths of the subintervals) $\{x_i-x_{i-1}\},$ and call the number $\parallel P\parallel$ the mesh size of the partition $P.$

A function $h:[a,b]\to C$ is called a step function if there exists a partition $P=\{x_0<x_1<...<x_n\}$ of $[a,b]$ and $n$ numbers $\{a_1,a_2,...,a_n\}$ such that $h\left(x\right)=a_i$ if $x_{i-1}<x<x_i.$ That is, $h$ is a step function if it is a constant function on each of the (open) subintervals $(x_{i-1},x_i)$ determined by a partition $P.$ Note that the values of a step function at the points $\left\{x_i\right\}$ of the partition are not restricted in any way.

A function $l:[a,b]\to R$ is called a polygonal function , or a piecewise linear function , if there exists a partition $P=\{x_0<x_1<...<x_n\}$ of $[a,b]$ and $n+1$ numbers $\{y_0,y_1,...,y_n\}$ such that for each $x\in [x_{i-1},x_i],$ $l\left(x\right)$ is given by the linear equation

$$l\left(x\right)=y_{i-1}+m_i(x-x_{i-1}),$$

where $m_i=(y_i-y_{i_1})/(x_i-x_{i-1}).$ That is, $l$ is a polygonal function if it is a linear function on each of the closed subintervals $[x_{i-1},x_i]$ determined by a partition $P.$ Note that the values of a piecewise linear function at the points $\left\{x_i\right\}$ of the partition $P$ are the same, whether we think of $x_i$ in the interval $[x_{i-1},x_i]$ or $[x_i,x_{i+1}].$ (Check the two formulas for $l\left(x_i\right).$ )

The graph of a piecewise linear function is the polygonal line joining the $n+1$ points $\left\{(x_i,y_i)\right\}.$

There is a natural generalization of the notion of a step function that works for any domain $S,$ e.g., a rectangle in the plane $C.$ Thus, if $S$ is a set, we define a partition of $S$ to be a finite collection $\{E_1,E_2,...,E_n\}$ of subsets of $S$ for which

1.   ${\cup }_{i=1}^n{E}_i=S,$ and
2.   $E_i\cap E_j=\varnothing$ if $i\ne j.$

Then, a step function on $S$ would be a function $h$ that is constant on each subset $E_i.$ We will encounter an even more elaborate generalized notion of a step function in Chapter V, but for now we will restrict our attention to step functions defined on intervals $[a,b].$

The set of polynomials and the set of step functions are both closed under additionand multiplication, and the set of rational functions is closed under addition,multiplication, and division.