330
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
Finally, if a continuous function of s E JR., say g, coincides with as on Q
i.e. if the relation f(s) = g(s) is satisfied for all SEQ, it must also do ~
for all s E JR., since every s E JR. is the limit of points of Q. Whence Theorem 1.
Let us venture a remark a propos the lemma above. Consider the function aX on the set Q and make x tend to an s E JR.; if one wanted to show
a priori that it converged to a limit - this would be another method of defhling as -, one would have to check that Cauchy's criterion of Chap. III, nO 10,
Theorem 13', held, in other words that, for all r > 0,
(2.11)
lax - aYI < r for x, y E Q sufficiently near to s.
This is precisely what the lemma assures: if u and v are chosen according to
the lemma, it is clear, since the function is increasing in Q, that (11) holds
so long as x, y E [u, v]. This argument could be used to prove directly the
existence in JR. of a unique continuous function equal to x ~ aX for x rational:
its value at s E JR. has to be the limit of aX when x E Q tends to s. It remains
to check that this limit is truly a continuous function of s, as the reader may
show easily.
3 - The calculus of real exponents
Given a number a > 0, the function f whose existence and the uniqueness
are assured by Theorem 1 is called the exponential function of base a. From
now on we write it in the usual form f(x) = aX, or in the form
for a reason which will soon appear. The meaning of the symbol aX may
appear obvious from the notation we have adopted, but in reality this is
justified only by the constructions and arguments of the preceding nO; there
is certainly nothing obvious in an expression such as 7r1l", the limit of the
sequence whose successive terms are
31
(31)1/10
7r' = 7r
,
3 14
(314) 1/100
t
7r'
= 7r
, e c.
It is no more obvious that the expression aX satisfies the index laws irresistibly
suggested to those who take their desires for reality or confuse causes with
effects: the rules of calculus explain the notation - introduced in the case
of rational exponents by the mathematicians of the XVIIth century, mainly
John Wallis and Newton -, and not conversely. One has to justify them in
this general framework.
Let us first prove the formula
(I)
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
Finally, if a continuous function of s E JR., say g, coincides with as on Q
i.e. if the relation f(s) = g(s) is satisfied for all SEQ, it must also do ~
for all s E JR., since every s E JR. is the limit of points of Q. Whence Theorem 1.
Let us venture a remark a propos the lemma above. Consider the function aX on the set Q and make x tend to an s E JR.; if one wanted to show
a priori that it converged to a limit - this would be another method of defhling as -, one would have to check that Cauchy's criterion of Chap. III, nO 10,
Theorem 13', held, in other words that, for all r > 0,
(2.11)
lax - aYI < r for x, y E Q sufficiently near to s.
This is precisely what the lemma assures: if u and v are chosen according to
the lemma, it is clear, since the function is increasing in Q, that (11) holds
so long as x, y E [u, v]. This argument could be used to prove directly the
existence in JR. of a unique continuous function equal to x ~ aX for x rational:
its value at s E JR. has to be the limit of aX when x E Q tends to s. It remains
to check that this limit is truly a continuous function of s, as the reader may
show easily.
3 - The calculus of real exponents
Given a number a > 0, the function f whose existence and the uniqueness
are assured by Theorem 1 is called the exponential function of base a. From
now on we write it in the usual form f(x) = aX, or in the form
for a reason which will soon appear. The meaning of the symbol aX may
appear obvious from the notation we have adopted, but in reality this is
justified only by the constructions and arguments of the preceding nO; there
is certainly nothing obvious in an expression such as 7r1l", the limit of the
sequence whose successive terms are
31
(31)1/10
7r' = 7r
,
3 14
(314) 1/100
t
7r'
= 7r
, e c.
It is no more obvious that the expression aX satisfies the index laws irresistibly
suggested to those who take their desires for reality or confuse causes with
effects: the rules of calculus explain the notation - introduced in the case
of rational exponents by the mathematicians of the XVIIth century, mainly
John Wallis and Newton -, and not conversely. One has to justify them in
this general framework.
Let us first prove the formula
(I)