11.7: Rational Functions and the Laplace Transform

Introduction

When dealing with operations on polynomials, the term rational function is a simple way to describe a particular relationship between two polynomials.

Definition: Rational Function

For any two polynomials, A and B, their quotient is called a rational function.

Example \(\PageIndex{1}\)

Below is a simple example of a basic rational function, \(f(x)\). Note that the numerator and denominator can be polynomials of any order, but the rational function is undefined when the denominator equals zero.

\[f(x)=\frac{x^{2}-4}{2 x^{2}+x-3} \label{11.24} \]

Properties of Rational Functions

In order to see what makes rational functions special, let us look at some of their basic properties and characteristics. If you are familiar with rational functions and basic algebraic properties, skip to the next section to see how rational functions are useful when dealing with the Laplace transform.

Roots

To understand many of the following characteristics of a rational function, one must begin by finding the roots of the rational function. In order to do this, let us factor both of the polynomials so that the roots can be easily determined. Like all polynomials, the roots will provide us with information on many key properties. The function below shows the results of factoring the above rational function, Equation \ref{11.24}.

\[f(x)=\frac{(x+2)(x-2)}{(2 x+3)(x-1)} \nonumber \]

Thus, the roots of the rational function are as follows:

Roots of the numerator are: \(\{-2,2\}\)

Roots of the denominator are: \(\{-3,1\}\)

Note

In order to understand rational functions, it is essential to know and understand the roots that make up the rational function.

Discontinuities

Because we are dealing with division of two polynomials, we must be aware of the values of the variable that will cause the denominator of our fraction to be zero. When this happens, the rational function becomes undefined, i.e. we have a discontinuity in the function. Because we have already solved for our roots, it is very easy to see when this occurs. When the variable in the denominator equals any of the roots of the denominator, the function becomes undefined.

Example \(\PageIndex{2}\)

Continuing to look at our rational function above, Equation \ref{11.24}, we can see that the function will have discontinuities at the following points: \(x=\{-3,1\}\)

In respect to the Cartesian plane, we say that the discontinuities are the values along the x-axis where the function is undefined. These discontinuities often appear as vertical asymptotes on the graph to represent the values where the function is undefined.

Domain

Using the roots that we found above, the domain of the rational function can be easily defined.

Definition: Domain

The group, or set, of values that are defined by a given function.

Example \(\PageIndex{3}\)

Using the rational function above, Equation \ref{11.24}, the domain can be defined as any real number \(x\) where \(x\) does not equal 1 or negative 3. Written out mathematically, we get the following:

\[\{x \in \mathbb{R} |(x \neq-3) \text { and }(x \neq 1)\} \nonumber \]

Intercepts

The x-intercept is defined as the point(s) where \(f(x)\), i.e. the output of the rational functions, equals zero. Because we have already found the roots of the equation this process is very simple. From algebra, we know that the output will be zero whenever the numerator of the rational function is equal to zero. Therefore, the function will have an x-intercept wherever \(x\) equals one of the roots of the numerator.

The y-intercept occurs whenever \(x\) equals zero. This can be found by setting all the values of \(x\) equal to zero and solving the rational function.

Rational Functions and the Laplace Transform

Rational functions often result when the Laplace transform is used to compute transfer functions for LTI systems. When using the Laplace transform to solve linear constant coefficient ordinary differential equations, partial fraction expansions of rational functions prove particularly useful. The roots of the polynomials in the numerator and denominator of the transfer function play an important role in describing system behavior. The roots of the polynomial in the numerator produce zeros of the transfer function where the system produces no output for an input of that complex frequency. The roots of the polynomial in the denominator produce poles of the transfer function where the system has natural frequencies of oscillation.

Summary

Once we have used our knowledge of rational functions to find its roots, we can manipulate a Laplace transform in a number of useful ways. We can apply this knowledge by representing an LTI system graphically through a pole-zero plot for analysis or design.