40
VIII – Cauchy Theory
compact convergence. This result covers every type of convergence encountered in practice, in particular L
p convergence considered above.
(v) Analyticity of holomorphic functions. Cauchy’s formula for a circle
assumes only that f is holomorphic in the sense initially defined in Chap. II,
n
◦ 19. Using the theory of Fourier series, it was directly shown in Chap. VII,
no 14 that in fact all holomorphic functions are analytic, i.e. have power series
expansions, and (10) was deduced. Cauchy’s formula provides under proof of
analyticity, that of Cauchy which everyone reproduced and which proceeds
in reverse order. It suffices to replace a by a variable z in (10), and to write
that
1/(ζ − z) = ζ
−1 /(1 − z/ζ) =
n ≥ 0
z
n ζ
−n−1 .
This geometric series expansion is justified as long as |z| < |ζ| = R. For given
z, this series of functions of ζ converges normally (Chap. III, n
◦ 8) on the
circle |ζ| = R since it is dominated by the series
q
n /R, with q = |z|/R < 1.
As the function f is continuous on the circle, it can be integrated term by
term. Hence
2πif (z) =
c n z
n
with c n =
f (ζ)ζ
−n−1 dζ ,
(4.19)
where integration is along the circle of radius R, qed.
Like the one of Chap. VII, this proof shows that expansion (19) holds
in the largest disc D centered at 0 contained in the domain G where f is
holomorphic. As indeed all circles centered at 0 are pairwise homotopic as
closed paths in the domain G − {0} where the function f (ζ)ζ
−n−1 being
integrated is holomorphic, the integral representing the c n is independent
from R as long as the closed disc |z| ≤ R is contained in G. Since the power
series then converges for |z| ≤ R, the result follows. This argument also shows
that Cauchy’s integral formula for a disc characterizes holomorphic functions
on the disc since it implies a power series expansion or else because the
function being integrated on a compact set depends holomorphically on the
parameter z, which allows theorems on differentiation under the
sign to be
applied.
(vi) Laurent Series. Consider a function f holomorphic on an annulus
G : r < |z| < R and let z be a point of G. Choose numbers r
and R
such that r < r
< |z| < R
< R and a ray D with initial point 0 which
does not pass through z; let A and B be the points where it meets circles
of radius r
and R
. Consider the closed path μ consisting of AB, followed
by the circumference |ζ| = R
oriented positively, then by BA followed in
turn by the circle |ζ| = r
oriented clockwise. It is obviously homotopic in
G − {z} to a circumference γ centered at z contained in G. The integrals of
VIII – Cauchy Theory
compact convergence. This result covers every type of convergence encountered in practice, in particular L
p convergence considered above.
(v) Analyticity of holomorphic functions. Cauchy’s formula for a circle
assumes only that f is holomorphic in the sense initially defined in Chap. II,
n
◦ 19. Using the theory of Fourier series, it was directly shown in Chap. VII,
no 14 that in fact all holomorphic functions are analytic, i.e. have power series
expansions, and (10) was deduced. Cauchy’s formula provides under proof of
analyticity, that of Cauchy which everyone reproduced and which proceeds
in reverse order. It suffices to replace a by a variable z in (10), and to write
that
1/(ζ − z) = ζ
−1 /(1 − z/ζ) =
n ≥ 0
z
n ζ
−n−1 .
This geometric series expansion is justified as long as |z| < |ζ| = R. For given
z, this series of functions of ζ converges normally (Chap. III, n
◦ 8) on the
circle |ζ| = R since it is dominated by the series
q
n /R, with q = |z|/R < 1.
As the function f is continuous on the circle, it can be integrated term by
term. Hence
2πif (z) =
c n z
n
with c n =
f (ζ)ζ
−n−1 dζ ,
(4.19)
where integration is along the circle of radius R, qed.
Like the one of Chap. VII, this proof shows that expansion (19) holds
in the largest disc D centered at 0 contained in the domain G where f is
holomorphic. As indeed all circles centered at 0 are pairwise homotopic as
closed paths in the domain G − {0} where the function f (ζ)ζ
−n−1 being
integrated is holomorphic, the integral representing the c n is independent
from R as long as the closed disc |z| ≤ R is contained in G. Since the power
series then converges for |z| ≤ R, the result follows. This argument also shows
that Cauchy’s integral formula for a disc characterizes holomorphic functions
on the disc since it implies a power series expansion or else because the
function being integrated on a compact set depends holomorphically on the
parameter z, which allows theorems on differentiation under the
sign to be
applied.
(vi) Laurent Series. Consider a function f holomorphic on an annulus
G : r < |z| < R and let z be a point of G. Choose numbers r
and R
such that r < r
< |z| < R
< R and a ray D with initial point 0 which
does not pass through z; let A and B be the points where it meets circles
of radius r
and R
. Consider the closed path μ consisting of AB, followed
by the circumference |ζ| = R
oriented positively, then by BA followed in
turn by the circle |ζ| = r
oriented clockwise. It is obviously homotopic in
G − {z} to a circumference γ centered at z contained in G. The integrals of