36
VIII – Cauchy Theory
corollary to Theorem 4, this is the product of 2πi and of the index of a with
respect to such a circle, which is obviously equal to 1. In view of the factor
f (a), the final result is, therefore, indeed 2πif (a). This is one of Cauchy’s
own proofs.
(iv) Modes of convergence of holomorphic functions. Recall that formula (10), an immediate consequence of the elementary theory of Fourier
series, is the essential tool in the proof of Weierstrass’ theorem on limits of
holomorphic functions (Chap. VII, § 4, no 19). Indeed, if f is holomorphic on
the open set G, if K is a compact subset of G and if r > 0 is strictly less
than the distance from K to the boundary of G, then, taking for μ the circle
|ζ − w| = r, (9) may be applied for all w ∈ K. Set ζ = w + re(t), whence
dζ = 2πire(t)dt and
f (ζ) =
n≥0
f
(n) (w)r
n e(t)
n /n! ,
Then,
f
(n) (w)/n! =
f (w + re(t)) r
−n e(t)
−n dt ,
(4.12)
where integration is over [0, 1]. Hence setting K(r) to be the compact set
contained in G, consisting of points at a distance ≤ r from K, and maximizing
of the left hand side over K, we get
f
(n)
K
≤ n!r
−n
f K(r) ,
(4.13)
which proves that, in the vector space of holomorphic functions on G, the map
f → f
(n) is continuous with respect to the topology of compact convergence.
25
Indeed, if a sequence (f p ) converges uniformly to a limit f on every compact
subset of G, the same holds for the successive derivatives of f in the real
sense, which is identical up to constant factors to the complex sense. The
limit function is, therefore, C
∞ and satisfies Cauchy’s condition by passing
to the limit. Its limit is, therefore, holomorphic, and as the derivatives in
the real sense converge uniformly to those of f on every compact subset, the
expected result follows (Chap. VII, no 19, Weierstrass’ theorem).
In fact, Weierstrass’ conclusion can be reached by making seemingly much
weaker assumptions than compact convergence on the convergence of the
sequence (f n ). Let us in particular consider L
p (1 ≤ p < +∞) convergence in
the theory of integration. It is defined by the norm
f p =
G
|f (z)|
p dm(z)
1/p
,
(4.14)
25 Recall (Chapter III, Appendix, no 8) that it is defined by the seminorms f →
f K , where K ⊂ G is an arbitrary compact set. The inequality obtained shows
that if f converges uniformly on K(r), then f
(n) converges uniformly on K.
VIII – Cauchy Theory
corollary to Theorem 4, this is the product of 2πi and of the index of a with
respect to such a circle, which is obviously equal to 1. In view of the factor
f (a), the final result is, therefore, indeed 2πif (a). This is one of Cauchy’s
own proofs.
(iv) Modes of convergence of holomorphic functions. Recall that formula (10), an immediate consequence of the elementary theory of Fourier
series, is the essential tool in the proof of Weierstrass’ theorem on limits of
holomorphic functions (Chap. VII, § 4, no 19). Indeed, if f is holomorphic on
the open set G, if K is a compact subset of G and if r > 0 is strictly less
than the distance from K to the boundary of G, then, taking for μ the circle
|ζ − w| = r, (9) may be applied for all w ∈ K. Set ζ = w + re(t), whence
dζ = 2πire(t)dt and
f (ζ) =
n≥0
f
(n) (w)r
n e(t)
n /n! ,
Then,
f
(n) (w)/n! =
f (w + re(t)) r
−n e(t)
−n dt ,
(4.12)
where integration is over [0, 1]. Hence setting K(r) to be the compact set
contained in G, consisting of points at a distance ≤ r from K, and maximizing
of the left hand side over K, we get
f
(n)
K
≤ n!r
−n
f K(r) ,
(4.13)
which proves that, in the vector space of holomorphic functions on G, the map
f → f
(n) is continuous with respect to the topology of compact convergence.
25
Indeed, if a sequence (f p ) converges uniformly to a limit f on every compact
subset of G, the same holds for the successive derivatives of f in the real
sense, which is identical up to constant factors to the complex sense. The
limit function is, therefore, C
∞ and satisfies Cauchy’s condition by passing
to the limit. Its limit is, therefore, holomorphic, and as the derivatives in
the real sense converge uniformly to those of f on every compact subset, the
expected result follows (Chap. VII, no 19, Weierstrass’ theorem).
In fact, Weierstrass’ conclusion can be reached by making seemingly much
weaker assumptions than compact convergence on the convergence of the
sequence (f n ). Let us in particular consider L
p (1 ≤ p < +∞) convergence in
the theory of integration. It is defined by the norm
f p =
G
|f (z)|
p dm(z)
1/p
,
(4.14)
25 Recall (Chapter III, Appendix, no 8) that it is defined by the seminorms f →
f K , where K ⊂ G is an arbitrary compact set. The inequality obtained shows
that if f converges uniformly on K(r), then f
(n) converges uniformly on K.