536 Chapter 8 • Infinite Series of Real Numbers
Recall that a real number x is called an algebraic number if it is a
solution of some polynomial equation
ao + a1x + a2x^2 + · · · + anxn = 0
with integer coefficients a 0 , a 1 , · · · , an. A transcendental number is a real
number that is not algebraic. For example, 97, Y13, and \/7 - ..yfl are alge-
braic, but 7r, e, e^100 , and sin 10 are transcendental.
Similarly, a function y = y(x) is an algebraic function if it is a solution
of some polynomial identity (true for all values of x)
ao(x) + a1(x)y + a2(x)y^2 + · · · + an(x)yn = 0,
where the coefficients ai(x) are polynomials with integer coefficients. A tran-
scendental function is a function that is not algebraic. At one time math-
ematicians thought that polynomial equations could be solved by explicit for-
mula using only "algebraic" methods (addition, subtraction, multiplication, di-
vision, and extraction of nth roots for integral n). The work of Abel and Galois
in the early nineteenth century proved this impossible in general. Nevertheless,
algebraic functions are still regarded as simpler than transcendental functions
since transcendental functions cannot be defined using polynomial equations.
Elementary functions are rational functions, ex, sin x, cos x, and all func-
tions that can be expressed as finite combinations of these and their inverses,
using algebraic operations, composition, and inverses. These include polynomi-
als, fractional powers, logarithms, all trigonometric functions and hyperbolic
functions, and their inverses. Virtually all functions encountered in beginning
calculus courses are elementary functions. Calling them elementary is just a
convention of no great importance. Nonelementary functions abound in math-
ematics. Indeed, applied mathematicians may use nonelementary functions so
routinely in their line of work that they find them as familiar as elementary
functions.
Of concern in this section is how elementary transcendental functions can
be defined and their values calculated. It is naive to believe, for example, that
23 ·^71 , log 10 56, sin 5.68, or err can be calculated from the "definitions" given
in elementary courses. We have devoted some space to this problem in earlier
chapters of this book. In Section 5.6 we gave rigorous definitions of exponen-
tials, powers, and logarithms using only continuity arguments. In Section 7. 7
we gave definitions of ln x, ex, and the trigonometric functions using the Rie-
mann integral. While these definitions are rigorous, they are not very useful in
computation. We shall now show that power series provide the most direct and
computationally useful definitions of the elementary transcendental functions.
We begin by asking you to imagine that you have never seen a definition
of exponential or logarithm functions or any of the trigonometric functions. In
what follows, you may use all the results of real analysis we have developed
so far except those related to these functions. We are going to define these