§2. Series expansions
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Note that, for s = 1, we obtain the formula
exp(z - z2/2 + ... ) = 1 + z, JzJ < 1.
This is proved even more easily: the two sides are analytic on D; they coincide
for real z since log(l + z) = z - z2/2 + ... for z E ] - 1,1[; by analytic
continuation, they are identical on D.
Second proof (z E lR, s E lR). This more elementary proof uses the functional equation for the exponential functions, but imposes restrictions on s
and z. Here one considers the series (1) as a function of s for given z E ]-1, 1 [,
and no longer as a function of z for s given.
Choose numbers z, sand t, complex for the moment, with JzJ < 1 to
ensure (Chap. II, nO 16) the absolute convergence of the series
(11.5)
(11.6)
1 + sz + s(s - 1)z2/2! + ... = L cn(s)zn,
1 + tz +t(t -1)z2/2! + ... = Lcn(t)zn.
For s, tEN, their sums are equal to (1 + Z)8 and (1 + z)t by the algebraic
binomial formula, so that, in this case,
(11.7)
We shall see that this formula persists for any complex s and t and z E C,
JzJ < 1. The argument which follows is due to Euler (1774).
Let us apply the multiplication rule for power series (Chap. II, nO 22).
We find an absolutely convergent power series LCn(S,t)zn with
(11.8)
Cn(s, t) = L ep(s)cp(t).
p+q=n
It thus reduces to proving the identity
(11.9)
Cn(s + t) = L cp(s)cq(t).
First of all it is clear that the coefficients figuring in (5) and (6) are polynomials in sand t respectively, so that the left hand side of (9) is a polynomial
in sand t. The relation (9) to be proved is thus an identity between two
polynomials in sand t, an identity which we know, by the algebraic binomial
formula, to be valid when one gives the variables s and t positive integer
values. By subtraction, it thus all reduces to proving the following general
result, where here one calls the variables x and y instead of sand t:
Principle of continuation of algebraic identities. Let