O'Reilly logo

Mathematics for Physicists by Graham Shaw, Brian R. Martin

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

2 Some basic functions and equations

The discussion of functions of one real variable is continued in this chapter by considering three classes of functions that play an important role in physical sciences. They are: simple algebraic functions; trigonometric functions; logarithms and exponentials.

2.1 Algebraic functions

Here we discuss polynomials and the more complicated functions that can be defined in terms of them.

2.1.1 Polynomials

The polynomial function has the general form

(2.1) Unnumbered Display Equation

where ai(i = 0, 1, 2, …, n) are constants, n is a non-negative integer (i.e. including zero), and the symbol Σ means that a sum is to be taken of all terms labelled by the indices 0, 1, 2, …, n. The value of n defines the order (or degree) of the polynomial. The expression x3 − 3x2 − 6x + 8 plotted in Figure 1.1 is therefore a polynomial of order 3.

The roots of polynomials are defined as the solutions of the equation

(2.2) Unnumbered Display Equation

and correspond to the points where a graph of Pn(x) crosses the x-axis. For first-order polynomials, (2.2) is a linear equation of the form

(2.3) Unnumbered Display Equation

where a and b are constants. This has one root, which is trivially given by x = −b/a.

For second-order polynomials, (

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required