370
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
Theorem 11 also provides an express proof of the relation exp(log x) = x
for x real. Put
whence
x = (1 + un/n)n.
The sequence (un) converges by definition to log x; Theorem 11 now assures
us that (1 + un/n)n tends to exp(logx), qed.
Theorem 11 enables us, and this is very useful, to determine all reasonable
functions with complex values that satisfy the addition formula
(13.5)
e(x + y) = e(x)e(y).
We have to distinguish two cases.
Corollary 1. Let e : lR ---+ C be a solution of (5), not identically zero,
possessing a derivative at x = O. Then
e(x) = exp(ex)
with c = e' (0).
To see this, we first note, once again, that (5) implies e(O) = 1, then that
e(nx) = e(x)n for n E N and so also
(13.6)
e(x/n)n = e(x).
By definition of the derivative, the quotient
e(x/n) - 1
=Xn
x/n
(13.7)
tends to a limit c. But (6) and (7) show that
Since TXn tends to ex, Theorem 11 shows that the right hand side tp.nds
to exp(ex); so it is equal to e(x) for all n, qed.
Another proof: since [e(x + h) - e(x)J/h = e(x)[e(h) - IJ/h, we see that
e(x) is everywhere differentiable and that e'(x) = ce(x) where c = e'(O). Now
the function exp(ex) possesses the same property. We deduce immediately
that the function e(x)/ exp(ex) has a zero derivative, so is constant and in
fact equal to 1 (put x = 0).
Corollary 2. Let e : C ---+ C be a solution of (5), not identically zero,
possessing a derivative in the complex sense at z = O. Then
(13.8)
e(z) = exp(cz)
with c = e'(O)
for all z E C.