§3. First concepts of analytic functions
183
whose coefficients are the sums 3 ~ l/w 4 , 5 ~ l/w 6 , •.. This formula is valid
for lui < R, i.e. in the largest open disc with centre 0 not containing any
nonzero period, but clearly not outside it because of the presence on the circumference lui = R of at least one period, i.e. of a pole of gJ.
One could expand the series (1) similarly in the same disc for k ~ 3. For
example
1
u- 3 - 2"w- 3 ~)n + 1)(n + 2)(u/w)n
l/u 3 + Lbnu n
with
1
bn = -2"(n + l)(n + 2) L l/wn+3 = -(n + l)an+t/2
by (8). This time bn = 0 for n even, in other words,
Comparing with (9), one sees that, for lui < R, one has
(23.10)
-2h(u) = gJ'(u),
the derived function of gJ( u) in the sense of the theory of analytic functions.
Since the derivative of the general term of (4) is -2/(u-w)3 by the traditional
rules of calculus, this result confirms that one can deduce (10) from (4) by
differentiating the series 66 which defines gJ(u) term-by-term with respect to
u. And since f and gJ' are analytic in the connected open set G = C - L, (10)
remains valid on all of G (nO 20).
However it may be, (10) allows us to show the periodicity of gJ(u)j this is
not obvious from its definition, in contrast to the case of h(u). (10) shows
that gJ' (u + w) = gJ' ( u ), so that the function
g(u) = gJ(u + w) - gJ(u),
analytic in G, has zero derivative. Taylor's formula then shows that it is
constant on a neighbourhood of each point of G, so on G (principle of analytic
continuation). One therefore has a relation
gJ(u + w) = gJ(u) + c(w)
66 We have shown in nO 19 that one can differentiate a power series term-by-term,
but here we are dealing with a series whose general term, though analytic, is not
a power ofu.