§2. Series expansions
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§ 2. Series expansions
9 - The number e. N apierian logarithms
In Chap. II, nO 22, we showed that the exponential series
(9.1)
exp{z) = 1 + z/l! + z2/2! + ... = L zn In! = L zlnj
satisfies the relation
(9.2)
exp{x + y) = exp{x). exp{y)
for all x, y E C and in particular for x and y real. It is clearly continuous
on JR (and even on C) since it is analytic. In view of Theorem 2 above, there
must exist a number e > 0 such that
(9.3)
for all x E JR, with necessarily
(9.4)
e = exp{l) = L l/n! = 1 + 1 + 1/2! + 1/3! + ....
Thus
(9.5)
where e is the number (4), an a priori miraculous result if one defines eX by
(I). The "miracle" stems from the addition formula for the series for expo
Formula (8.1) can be then be written
(9.6)
exp'{x) = log{e).expx.
But the formula in Chap. II, nO 19, for differentiating power series, namely
that
shows that
(9.7)
exp'(x) = expx
as we have already said N times. It follows that
(9.8)
loge = 1.
And since the function log x = lim n(xl/n -1) satisfies the functional equation
log(xy) = logx+logy (Chap. II, n° 10), is increasing and not identically zero,
there is a number a > 0 such that logx = loga x (Theorem 5), namely the
number such that loga = 1. In other words, we see that Napier's function
log is simply loge' i.e. the inverse map of eXPe = exp: