48
VIII – Cauchy Theory
f
(z)/f (z) = p/(z − a) + g
(z)/g(z) .
The function g
/g being holomorphic at a, Res(f
/f, a) = p = v a (f ) and the
residue formula then shows that
μ
f
(z)
f (z)
dz = 2πi
Ind μ (a)v a (f ) = 2πiv μ (f ) ,
(5.11)
where the sum, extended to all a ∈ G, only has a finite number of non-zero
terms. If for example f is holomorphic everywhere on G and if μ is a simple
closed curve whose interior is contained in G and on which f does not vanish,
then integral (11) allows us to calculate the total number of zeros of f in the
interior of μ; this number takes into account the order or multiplicity v a (f )
of each zero.
As a function of f , the left hand side of (11) satisfies a remarkable continuity property resulting from that of the map f → f
(n) studied above. The
open set G and the path μ being fixed, first note that if f does not vanish
on Supp(μ), then neither do functions g defined on G and sufficiently near
f in the sense of compact convergence. Then choose a number r > 0 strictly
less that the distance from Supp(μ) to the boundary of G and let K ⊂ G be
the compact set of points whose distance to Supp(μ) is ≤ r. Since f does not
vanish on Supp(μ), for sufficiently small r, it does not vanish on K either.
The lower bound d of |f (z)| in K is, therefore, > 0 if r is sufficiently small,
and that of |g(z)| is ≥ d/2 if f − g K < d/2.If g is holomorphic, the results
stated in n
◦ 4, (iv) show that f
− g
μ(I) ≤ Md where M does not depend
on g. If f − g K is sufficiently small, the integral
2πiv μ (g) − 2πiv μ (f ) =
μ
g
(z)
g(z)
−
f
(z)
f (z)
dz
is well defined and considering an upper bound for it,
33 it follows that for
f − g K sufficiently small, |v μ (g) − v μ (f )| < 1, and so = 0. As a result, the
number v μ (f ) of roots of a holomorphic function f in the interior of a closed
path μ is a continuous function of f with respect to the topology of compact
convergence. In particular, for every functions f holomorphic on G and all
a ∈ G, there exist r > 0, ρ > 0 and a compact set K ⊂ G such that the
number of zeros in the disc |z − a| < r of any function g holomorphic on G
and satisfying g − f K < ρ is the same as that of f .
If f and g are replaced by f − c and f − c
, where c and c
are constants,
so that g − f K = |c
− c|, then, for given c, the equations f (z) = c and
f (z) = c
have the same number of solutions in the interior of μ, provided
|c
− c| is sufficiently small. A direct argument : if f (z) − c does not vanish on
33 It all amounts to finding an upper bound for an expression of the form |f/g−p/q|
when lower bounds > 0 of g and q and upper bounds for f , p, |f − p| and |g − q|
are known . See rules of Chapter III, § 2, n
◦ 7 with respect to algebraic operations
on uniformly convergent sequences.
VIII – Cauchy Theory
f
(z)/f (z) = p/(z − a) + g
(z)/g(z) .
The function g
/g being holomorphic at a, Res(f
/f, a) = p = v a (f ) and the
residue formula then shows that
μ
f
(z)
f (z)
dz = 2πi
Ind μ (a)v a (f ) = 2πiv μ (f ) ,
(5.11)
where the sum, extended to all a ∈ G, only has a finite number of non-zero
terms. If for example f is holomorphic everywhere on G and if μ is a simple
closed curve whose interior is contained in G and on which f does not vanish,
then integral (11) allows us to calculate the total number of zeros of f in the
interior of μ; this number takes into account the order or multiplicity v a (f )
of each zero.
As a function of f , the left hand side of (11) satisfies a remarkable continuity property resulting from that of the map f → f
(n) studied above. The
open set G and the path μ being fixed, first note that if f does not vanish
on Supp(μ), then neither do functions g defined on G and sufficiently near
f in the sense of compact convergence. Then choose a number r > 0 strictly
less that the distance from Supp(μ) to the boundary of G and let K ⊂ G be
the compact set of points whose distance to Supp(μ) is ≤ r. Since f does not
vanish on Supp(μ), for sufficiently small r, it does not vanish on K either.
The lower bound d of |f (z)| in K is, therefore, > 0 if r is sufficiently small,
and that of |g(z)| is ≥ d/2 if f − g K < d/2.If g is holomorphic, the results
stated in n
◦ 4, (iv) show that f
− g
μ(I) ≤ Md where M does not depend
on g. If f − g K is sufficiently small, the integral
2πiv μ (g) − 2πiv μ (f ) =
μ
g
(z)
g(z)
−
f
(z)
f (z)
dz
is well defined and considering an upper bound for it,
33 it follows that for
f − g K sufficiently small, |v μ (g) − v μ (f )| < 1, and so = 0. As a result, the
number v μ (f ) of roots of a holomorphic function f in the interior of a closed
path μ is a continuous function of f with respect to the topology of compact
convergence. In particular, for every functions f holomorphic on G and all
a ∈ G, there exist r > 0, ρ > 0 and a compact set K ⊂ G such that the
number of zeros in the disc |z − a| < r of any function g holomorphic on G
and satisfying g − f K < ρ is the same as that of f .
If f and g are replaced by f − c and f − c
, where c and c
are constants,
so that g − f K = |c
− c|, then, for given c, the equations f (z) = c and
f (z) = c
have the same number of solutions in the interior of μ, provided
|c
− c| is sufficiently small. A direct argument : if f (z) − c does not vanish on
33 It all amounts to finding an upper bound for an expression of the form |f/g−p/q|
when lower bounds > 0 of g and q and upper bounds for f , p, |f − p| and |g − q|
are known . See rules of Chapter III, § 2, n
◦ 7 with respect to algebraic operations
on uniformly convergent sequences.