158
II - Convergence: Discrete variables
f(a) = J'(a) = ... = fP-l(a) = 0, f(p)(a) f= o.
One then says that a is a zero of order p of f. The situation is the opposite
to that of the case (i): there exists an open disc with centre a in which a is
the only zero of f (principle of isolated zeros). Indeed
(19.20)
f(a + h) = f(p) (a)h[Pl [1+?h+?h2 + ... J
with scalar coefficients denoted by ? since their values are of little importance.
For Ihl small, the sum ?h+ ... is, in modulus, :::; Mlhl with a constant M > 0,
since it is the product of h by a convergent power series. Since there exists a
number r > 0 such that one has Mlhl < 1/2 for Ihl < r, the sum between the
brackets [ J is in modulus> 1/2 for Ihl < r, so that f(a + h) cannot vanish
in the disc D(a,r), except at the point a, qed.
An immediate consequence of this result is that if mutually distinct zeros
of f tend to a point a E G, then f vanishes on a neighbourhood of a, since
any disc with centre a then contains infinitely many zeros of f.
20 - The principle of analytic continuation 50
Case (i) above leads to an a priori curious result: if all the derivatives of f
vanish at a, then the function f is identically zero on G - and not only on
a neighbourhood of a - subject to a "connectedness" hypothesis on G. This
"rigidity" property of analytic functions is quite special to them; in lR. (or in
any Cartesian space), a function as "smooth" as one wants, i.e. possessing
derivatives of arbitrarily high order with respect to the real variables on which
it depends, can very well vanish on an open set without being identically
zero: this is the case for f(x) = exp( -1/x 2 ) for x > 0, = 0 for x :::; 0 (the
existence of all the successive derivatives at 0 is not obvious; see Chap. IV,
nO 5). Equivalent formulation: if two analytic functions f and g on an open
connected G coincide on the neighbourhood of a point a of G or, it comes to
the same, have equal successive derivatives at a, then they are equal on all
ofG.
For let f be an analytic function on G and suppose that f and all its
derivatives vanish at a E G; then, by Taylor's formula, f vanishes on a neighbourhood of a. Let bEG and first assume that the line segment [a, bJ lies
entirely in G. We shall show that fez) = 0 on a neighbourhood of b also. The
segment [a, bJ is the set of complex numbers of the form
z(t) = a + t(b - a) = (1 - t)a + tb,
0:::; t :::; 1,
the points a and b corresponding to t = 0 and t = 1. Let E C [O,IJ = I be
the set of t possessing the following property: the function f vanishes on a
50 This nO will be called on only sporadically before Chap. VII and is not indispensable for the moment.
II - Convergence: Discrete variables
f(a) = J'(a) = ... = fP-l(a) = 0, f(p)(a) f= o.
One then says that a is a zero of order p of f. The situation is the opposite
to that of the case (i): there exists an open disc with centre a in which a is
the only zero of f (principle of isolated zeros). Indeed
(19.20)
f(a + h) = f(p) (a)h[Pl [1+?h+?h2 + ... J
with scalar coefficients denoted by ? since their values are of little importance.
For Ihl small, the sum ?h+ ... is, in modulus, :::; Mlhl with a constant M > 0,
since it is the product of h by a convergent power series. Since there exists a
number r > 0 such that one has Mlhl < 1/2 for Ihl < r, the sum between the
brackets [ J is in modulus> 1/2 for Ihl < r, so that f(a + h) cannot vanish
in the disc D(a,r), except at the point a, qed.
An immediate consequence of this result is that if mutually distinct zeros
of f tend to a point a E G, then f vanishes on a neighbourhood of a, since
any disc with centre a then contains infinitely many zeros of f.
20 - The principle of analytic continuation 50
Case (i) above leads to an a priori curious result: if all the derivatives of f
vanish at a, then the function f is identically zero on G - and not only on
a neighbourhood of a - subject to a "connectedness" hypothesis on G. This
"rigidity" property of analytic functions is quite special to them; in lR. (or in
any Cartesian space), a function as "smooth" as one wants, i.e. possessing
derivatives of arbitrarily high order with respect to the real variables on which
it depends, can very well vanish on an open set without being identically
zero: this is the case for f(x) = exp( -1/x 2 ) for x > 0, = 0 for x :::; 0 (the
existence of all the successive derivatives at 0 is not obvious; see Chap. IV,
nO 5). Equivalent formulation: if two analytic functions f and g on an open
connected G coincide on the neighbourhood of a point a of G or, it comes to
the same, have equal successive derivatives at a, then they are equal on all
ofG.
For let f be an analytic function on G and suppose that f and all its
derivatives vanish at a E G; then, by Taylor's formula, f vanishes on a neighbourhood of a. Let bEG and first assume that the line segment [a, bJ lies
entirely in G. We shall show that fez) = 0 on a neighbourhood of b also. The
segment [a, bJ is the set of complex numbers of the form
z(t) = a + t(b - a) = (1 - t)a + tb,
0:::; t :::; 1,
the points a and b corresponding to t = 0 and t = 1. Let E C [O,IJ = I be
the set of t possessing the following property: the function f vanishes on a
50 This nO will be called on only sporadically before Chap. VII and is not indispensable for the moment.