§ 1. Direct construction
339
Choose an a > 0 different from 1 and put g(x) = eXPa[f(x)], in other
words g = eXPa of. Then
g(xy) = eXPa[f(x) + fey)] = eXPa[f(x)]. eXPa[f(y)] = g(x)g(y).
Like the functions f and eXPa' the function g is continuous or monotone. It
is therefore a power function by the preceding theorem, in other words, there
exists an s E JR. such that af(x) = x 8 for all x > 0, whence
But every function proportional to a logarithmic function is itself of the same
type as we saw at the end of nO 3, whence Theorem 5.
Theorem 5 applies clearly, and above all, to the function
log x = lim n (xl/n - 1)
of Chap. II, nO 10, Theorem 3, and Theorem 2 applies to the function
exp x = L xn In! for which we established the addition formula in Chap. II,
nO 22; these functions are indeed continuous and monotone. Compare with
Chap. III, n° 2, example 1, where we proved in the same way that log[exp(x)]
is proportional to x. We shall return to this in the second part of this chapter, but may note now that if exp(x) = aX, then necessarily a = exp(l) =
e = L lin!. And if one knows that log[exp(x)] = x, as we have shown (same
reference), then log = loge.
7 - Derivatives of the exponential functions: direct method
Let f(x) = eXPa x = aX be an exponential function. Suppose that we have
established the existence of 1'(0). Then, for all x E JR.,
f(x+h)
f(x)f(h) = f(x)[f(O) + j'(O)h + o(h)]
f(x) + f(x)j'(O)h + o(h)
as h --+ 0, whence the existence of 1'(x) = c.f(x), with c = 1'(0); from
this it is clear that f has successive derivatives of all orders, and that
f{p)(x) = cp.f(x) for any p, so this function is indefinitely differentiable in
the sense of Chap. III, nO 15.
We therefore have to show that, for all a > 0, the ratio
(7.1)
(a h - l)/h
tends to a limit when h tends to o. We restrict to the case where a > 1, as
the other case can be treated similarly (or reduced to it).
339
Choose an a > 0 different from 1 and put g(x) = eXPa[f(x)], in other
words g = eXPa of. Then
g(xy) = eXPa[f(x) + fey)] = eXPa[f(x)]. eXPa[f(y)] = g(x)g(y).
Like the functions f and eXPa' the function g is continuous or monotone. It
is therefore a power function by the preceding theorem, in other words, there
exists an s E JR. such that af(x) = x 8 for all x > 0, whence
But every function proportional to a logarithmic function is itself of the same
type as we saw at the end of nO 3, whence Theorem 5.
Theorem 5 applies clearly, and above all, to the function
log x = lim n (xl/n - 1)
of Chap. II, nO 10, Theorem 3, and Theorem 2 applies to the function
exp x = L xn In! for which we established the addition formula in Chap. II,
nO 22; these functions are indeed continuous and monotone. Compare with
Chap. III, n° 2, example 1, where we proved in the same way that log[exp(x)]
is proportional to x. We shall return to this in the second part of this chapter, but may note now that if exp(x) = aX, then necessarily a = exp(l) =
e = L lin!. And if one knows that log[exp(x)] = x, as we have shown (same
reference), then log = loge.
7 - Derivatives of the exponential functions: direct method
Let f(x) = eXPa x = aX be an exponential function. Suppose that we have
established the existence of 1'(0). Then, for all x E JR.,
f(x+h)
f(x)f(h) = f(x)[f(O) + j'(O)h + o(h)]
f(x) + f(x)j'(O)h + o(h)
as h --+ 0, whence the existence of 1'(x) = c.f(x), with c = 1'(0); from
this it is clear that f has successive derivatives of all orders, and that
f{p)(x) = cp.f(x) for any p, so this function is indefinitely differentiable in
the sense of Chap. III, nO 15.
We therefore have to show that, for all a > 0, the ratio
(7.1)
(a h - l)/h
tends to a limit when h tends to o. We restrict to the case where a > 1, as
the other case can be treated similarly (or reduced to it).