160
II - Convergence: Discrete variables
To sum up, we have partitioned the open set G as two disjoint open sets
G(a) and G'(a), the first being nonempty since a E G(a). When G(a) = G,
one says that the open set G is connected. One can then, without leaving G,
join any two points band c of G by a broken line: one goes first from b to the
point a then from this to the point c. It is clear that this property characterises
the open connected sets: one has G(a) = G for any a E G. In the general
case, G(a), which is clearly connected, is called the connectedness component
of a in G or, if the point a is not explicit, a connectedness component of G.
Figure 7 shows an open set G possessing three connectedness components.
One can also define an open connected set as follows: it is impossible to
partition G as two open (disjoint, as in every partition) nonempty sets.
In order to formulate these results conveniently, it is useful to introduce
the concept of an isolated point of a subset of CC: this means that there exists
an open disc having this point as its centre and not containing any other
point of the set than the point considered; we shall meet this concept again
in Chap. III a propos limits. In particular this allows us to define the isolated
zeros of an analytic function f: these are the isolated points of the set Z(f)
of zeros of f, i.e. the points a possessing the following property: there exists
an r > 0 such that
Iz - al < r & f(z) = 0 <====> z = a.
It is now clear that at the end of the preceding nO we proved the following
result:
Theorem 15. Let f be an analytic function on an open subset G of C and
let a be a zero of f in G. Then either a is an isolated zero of f, or else f
vanishes on the connectedness component of a in G.
An equivalent formulation assuming G connected: either all the points of
Z(f) are isolated, or else f = O. If the function f(z) = Re(z) were analytic
in C, it would be identically zero since its zeros are manifestly not isolated.
Note that the zeros of an analytic function in an open connected G subset of
C may a.ll the same accumulate at the frontier of G; example: G = C - {O},
f(z) = sin(l/z), which vanishes at all the points l/mr, which converge to
o ¢. G. There are functions f defined and analytic for I z I < 1 and such that
there are infinitely many zeros of f in the neighbourhood of every point of
the boundary circle Izl = 1 (the "automorphic functions" of Henri Poincare
for example).
Corollary (Principle of analytic continuation). Let f and 9 be two
analytic functions on an open connected subset G of C. Suppose that there
are mutually distinct points an of G converging to a limit a E G and such
that
f(an ) = g(an )
for all n. Then f(z) = g(z) for any z E G.
II - Convergence: Discrete variables
To sum up, we have partitioned the open set G as two disjoint open sets
G(a) and G'(a), the first being nonempty since a E G(a). When G(a) = G,
one says that the open set G is connected. One can then, without leaving G,
join any two points band c of G by a broken line: one goes first from b to the
point a then from this to the point c. It is clear that this property characterises
the open connected sets: one has G(a) = G for any a E G. In the general
case, G(a), which is clearly connected, is called the connectedness component
of a in G or, if the point a is not explicit, a connectedness component of G.
Figure 7 shows an open set G possessing three connectedness components.
One can also define an open connected set as follows: it is impossible to
partition G as two open (disjoint, as in every partition) nonempty sets.
In order to formulate these results conveniently, it is useful to introduce
the concept of an isolated point of a subset of CC: this means that there exists
an open disc having this point as its centre and not containing any other
point of the set than the point considered; we shall meet this concept again
in Chap. III a propos limits. In particular this allows us to define the isolated
zeros of an analytic function f: these are the isolated points of the set Z(f)
of zeros of f, i.e. the points a possessing the following property: there exists
an r > 0 such that
Iz - al < r & f(z) = 0 <====> z = a.
It is now clear that at the end of the preceding nO we proved the following
result:
Theorem 15. Let f be an analytic function on an open subset G of C and
let a be a zero of f in G. Then either a is an isolated zero of f, or else f
vanishes on the connectedness component of a in G.
An equivalent formulation assuming G connected: either all the points of
Z(f) are isolated, or else f = O. If the function f(z) = Re(z) were analytic
in C, it would be identically zero since its zeros are manifestly not isolated.
Note that the zeros of an analytic function in an open connected G subset of
C may a.ll the same accumulate at the frontier of G; example: G = C - {O},
f(z) = sin(l/z), which vanishes at all the points l/mr, which converge to
o ¢. G. There are functions f defined and analytic for I z I < 1 and such that
there are infinitely many zeros of f in the neighbourhood of every point of
the boundary circle Izl = 1 (the "automorphic functions" of Henri Poincare
for example).
Corollary (Principle of analytic continuation). Let f and 9 be two
analytic functions on an open connected subset G of C. Suppose that there
are mutually distinct points an of G converging to a limit a E G and such
that
f(an ) = g(an )
for all n. Then f(z) = g(z) for any z E G.