422 Chapter 7 • The Riemann Integral
- 7 *Elementary Transcendental Functions
In this section we define the exponential,^15 logarithmic,^15
and trigonometric functions. We call for rigorous proofs of
their properties, well known from calculus and used routinely
throughout real analysis. The section can be covered lightly in
class, assigned as an independent reading project, or omitted
completely. There is no exercise set in this section; instead,
you are asked to fill in the proofs of results stated but left
unproved in the text.
For alternative approaches, see Sections 5.6 and 8.8.You are surely familiar with the definition of ex as the inverse of the function
ln x, which is defined in elementary calculus courses by an integral. But you may
be unfamiliar with the definition of sin x as the inverse of a function defined by
an integral. Indeed, you may be intrigued by the fact that in calculus courses
we seem to accept the trigonometric functions without definition. That is, we
seem to assume that they are defined somewhere outside of calculus. We are
now about to remedy that situation.
In this section we show that, besides providing definitions of the logarithmic
and exponential functions, the Riemann integral enables us to give rigorous
definitions of the trigonometric functions. We shall use the integral to define
these functions and outline proofs of their fundamental properties. We call these
functions "transcendental,'' because their values cannot be calculated as roots
of polynomial equations with rational coeffi.cients.^16
THE NATURAL LOGARITHM FUNCTIONDefinition 7.7.1 The natural logarithm function lnx : (0, oo) ---+JR is de-
fined by
lnx = Jx 1 ldt t (for x > 0).
Remarks 7.7.2 (a) lnx exists for all x > 0.
(b) ln x < 0 if 0 < x < 1; ln x = 0 if x = 1; ln x > 0 if x > 1.
(c)
(d)
ln x is continuous and strictly increasing on (0, oo ).
lnx is differentiable on (O, oo), and dd lnx = ~-
x x- The exponential and logarithmic functions were defined differently in Section 5.6. This
section is independent of that one, but the definitions are shown to be equivalent. - See Exercise 2.8.17 and the concluding remarks in Section 7.6.