§2. Series expansions
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as one sees on differentiating the power series term-by-term.
Note finally that with this definition of complex powers, we have
exp(z) = e Z for all z E C,
which often allows us to simplify the notation.
11 - Newton's binomial series
Theorem 8. We have
(11.1)
Ns(z) = 1 + sz + s(s - l)z2/2! + ... =
L s(s - 1) ... (s - n + l)z[n]
Jor -1 < z < 1 and sEC.
Since
10g(1 + z) = z - z2/2 + z3/3 - ...
for z E ] - 1, 1 [ and as = exp( s. log a) (a theorem for real s, the definition
of as for s complex and a > 0), Theorem 8 becomes a particular case of the
following result:
Theorem 8 bis. We have
(11.2) exp [s(z - z2/2 + z3/3 - ... )] = L s(s - 1) ... (s - n + l)z[n]
n;:::O
Jor s,z E C and Izl < 1.
There are several methods of proving these theorems, all of them instructive. Let us start with the best, which presents the drawback or the advantage
- everything depends on the point of view ... - of using the general theorems
of Chap. II, n° 19 on complex-analytic functions, and settles the question
in a way that was probably beyond the scope of anyone before Weip.rstrass'
systematic study of analytic functions.
First proof (general case). Consider the two sides of (2) as functions of z
with sEC given. These are analytic functions on the disc D : Izl < 1.
Consider first the left hand side J(z) = exp[sL(z)]. On putting 13
(11.3)
L(z) = z - z2/2 + ... ,
L is analytic on D and exp is analytic on C, so the composition of these two
functions is analytic (Chap. II, n° 22, Theorem 17).
13 We hold back from writing log(l + z), an expression which we have not yet
defined for z complex and which, when we do so later, may take infinitely many
values.