IV - Powers, Exponentials, Logarithms,
Trigonometric Functions
§1. Direct construction
§2. Expansions in series
'§9. Infinite products - §4. The topology of the functions Arg(z)
and .cog z
The general theorems of Chapters II and III have allowed us, in passing, to
~a.blish most of the principal properties of the elementary functions which
crop up everywhere in analysis. In this chapter we shall go over it all systematically. One can do this in various ways, each as instructive as the other. A first
method (nO 1 to 8) consists of erecting the theory from a minimum of knowledge, in particular using neither the theory of power series nor the function
exp, particularly its addition formula. A second, contrasting, method, starts
from these and goes much further. In particular it allows us to construct
a. rigorous analytic theory of the trigonometric functions. A third method
would be to define the function log x as an integral and from this deduce its
properties as well as those of the exponential functions.
§1. Direct construction
1 - Rational exponents
The first problem to resolve is to define the expression X S for every number
x> 0 (strict inequality) and every real (and even, later, complex) exponent s.
The construction is effected in several stages: first one treats the case where
8 E Z, then the case where SEQ and finally the general case. Naturally
we want eventually to obtain strictly positive expressions X S satisfying the
. "obvious" rules
(1.1)
valid when the exponents are integers of any sign, a case which we assume
familiar to the reader.
If S = p/q is rational, the rules (1) entail
(1.2)
(xp/q)q = x p .
Trigonometric Functions
§1. Direct construction
§2. Expansions in series
'§9. Infinite products - §4. The topology of the functions Arg(z)
and .cog z
The general theorems of Chapters II and III have allowed us, in passing, to
~a.blish most of the principal properties of the elementary functions which
crop up everywhere in analysis. In this chapter we shall go over it all systematically. One can do this in various ways, each as instructive as the other. A first
method (nO 1 to 8) consists of erecting the theory from a minimum of knowledge, in particular using neither the theory of power series nor the function
exp, particularly its addition formula. A second, contrasting, method, starts
from these and goes much further. In particular it allows us to construct
a. rigorous analytic theory of the trigonometric functions. A third method
would be to define the function log x as an integral and from this deduce its
properties as well as those of the exponential functions.
§1. Direct construction
1 - Rational exponents
The first problem to resolve is to define the expression X S for every number
x> 0 (strict inequality) and every real (and even, later, complex) exponent s.
The construction is effected in several stages: first one treats the case where
8 E Z, then the case where SEQ and finally the general case. Naturally
we want eventually to obtain strictly positive expressions X S satisfying the
. "obvious" rules
(1.1)
valid when the exponents are integers of any sign, a case which we assume
familiar to the reader.
If S = p/q is rational, the rules (1) entail
(1.2)
(xp/q)q = x p .