§3. First concepts of analytic functions
161
It is clear that then the point a is a non-isolated zero of the function f - g.
It is important that the limit must belong to G.
Theorem 15 is trivially false for open nonconnected sets: take for G the
union of two disjoint open discs and for f the function equal to 0 on the first
and to 1 on the second.
Although the preceding results concern functions defined on an open subset of C, they also apply to a class of functions defined on an interval I of ~.
Such a function f, with possibly complex values, is said to be real-analytic
(or analytic for short if no confusion with the complex-analytic functions of
nO 19 is possible) if, for all a E I, it can be represented on an interval with
centre a by a power series in x - aj this is the same definition as in the case
of C, except that the variable takes only real values.
In truth, there is hardly any difference between the two concepts: for a
function f(x) defined in an interval I c ~ to be real-analytic it is necessary
and sufficient that there exists an open subset G in C containing I and a
function g(z) defined and complex-analytic in G such that
f(x) = g(x) for all x E I.
This condition is clearly sufficient. To show the necessity, one starts from
the fact that, for all a E I, there exists a convergent power series ga(z) =
I:en(a)(z - a)n and an open disc D(a) C C with centre a such that (i)
ga converges in D(a), and possibly elsewhere, (ii) ga(x) = f(x) for all x E
In D(a). Now let G be the union of all the open discs D(a). This is clearly
an open subset of C. It is connected: if u E D(a) and v E D(b), one can join
u to v in G by following the ray [ua], the segment lab] of I and the ray [bv]
of D(b). If, moreover, D(a) and D(b) intersect, then we have
(20.1)
in InD(a) nD(b), an interval of~ not reducing to a point. Since D(a) nD(b)
is connected the principle of analytic continuation then shows that (1) is true
in all of D(a) n D(b).
Having done this one can define the function 9 on G by putting g(z) =
9a(Z) for all a E I such that z E D(a). Even though the point a can be chosen
in many different ways for a given z, this definition is not ambiguous because
of51 (1). It is clear that 9 is complex-analytic in each D(a), so on G, and that
9 = f on I, qed.
51 More generally: let E be a set, and (Ei) any family of subsets of E having union
E, and, for each i, let gi be a function with arbitrary values defined on E i . For
there to exist a function 9 on E such that 9 = gi on E. for all i, it is necessary
and sufficient that gi = gj in Ei n Ej for all i and j. Chap. I, §1.5.
161
It is clear that then the point a is a non-isolated zero of the function f - g.
It is important that the limit must belong to G.
Theorem 15 is trivially false for open nonconnected sets: take for G the
union of two disjoint open discs and for f the function equal to 0 on the first
and to 1 on the second.
Although the preceding results concern functions defined on an open subset of C, they also apply to a class of functions defined on an interval I of ~.
Such a function f, with possibly complex values, is said to be real-analytic
(or analytic for short if no confusion with the complex-analytic functions of
nO 19 is possible) if, for all a E I, it can be represented on an interval with
centre a by a power series in x - aj this is the same definition as in the case
of C, except that the variable takes only real values.
In truth, there is hardly any difference between the two concepts: for a
function f(x) defined in an interval I c ~ to be real-analytic it is necessary
and sufficient that there exists an open subset G in C containing I and a
function g(z) defined and complex-analytic in G such that
f(x) = g(x) for all x E I.
This condition is clearly sufficient. To show the necessity, one starts from
the fact that, for all a E I, there exists a convergent power series ga(z) =
I:en(a)(z - a)n and an open disc D(a) C C with centre a such that (i)
ga converges in D(a), and possibly elsewhere, (ii) ga(x) = f(x) for all x E
In D(a). Now let G be the union of all the open discs D(a). This is clearly
an open subset of C. It is connected: if u E D(a) and v E D(b), one can join
u to v in G by following the ray [ua], the segment lab] of I and the ray [bv]
of D(b). If, moreover, D(a) and D(b) intersect, then we have
(20.1)
in InD(a) nD(b), an interval of~ not reducing to a point. Since D(a) nD(b)
is connected the principle of analytic continuation then shows that (1) is true
in all of D(a) n D(b).
Having done this one can define the function 9 on G by putting g(z) =
9a(Z) for all a E I such that z E D(a). Even though the point a can be chosen
in many different ways for a given z, this definition is not ambiguous because
of51 (1). It is clear that 9 is complex-analytic in each D(a), so on G, and that
9 = f on I, qed.
51 More generally: let E be a set, and (Ei) any family of subsets of E having union
E, and, for each i, let gi be a function with arbitrary values defined on E i . For
there to exist a function 9 on E such that 9 = gi on E. for all i, it is necessary
and sufficient that gi = gj in Ei n Ej for all i and j. Chap. I, §1.5.