42
VIII – Cauchy Theory
Hence formula (13) gives a series expansion in the annulus r
< |z| < R
f (z) =
Z
c n z
n
(4.21)
whose coefficients are obtained by integrating f (ζ)ζ
−n−1 either over |ζ| = R
,
or over |ζ| = r
. The choice of the circle is in fact unimportant provided it is
in the interior of the annulus G : r < |z| < R since all circles are obviously
homotopic in G as closed paths. And as the expansion holds for r
< |z| < R
provided that r < r
< R
< R, this means that it holds in all of G. This is
Laurent’s theorem with coefficients given by an integral.
5 – The Residue Formula
(i) The residue formula. A formula (4.10) valid for any closed path μ in G,
any function f holomorphic on G and any z ∈ G, as well as a more general
and extremely classical result, can be proved in a simply connected domain :
Cauchy’s residue formula, an inexhaustible source of exercises and examination questions that, though sometimes subtle, have long become stale. Indeed,
everything can be proved at the same time and G need not be assumed to
be simply connected provided μ is taken to be null-homotopic in G.
First some remarks about the Laurent series of a function f holomorphic
for 0 < |z − a| < R, a being a singular isolated point of f (Chapter VII, § 4,
n
◦ 16). We saw above that it is given by
f (z) =
c n (z − a)
n
with 2πic n =
f (ζ)(ζ − a)
−n−1 dζ
(5.1)
where integration is over any circle t → a + r. exp(2πit) of radius r < R
centered at a. The sum of the terms of degree n < 0, i.e. the polar or singular
part of the Laurent series, is a power series in w = 1/(z − a); it converges
for 0 < |z − a| < R, i.e. for |w| > 1/R; Now, the domain of convergence of
a power series is the interior of a disc; if it converges outside a disc, then it
converges everywhere. Therefore, the series given by the terms of negative
degree in (1) converges for all z = a. So, in the neighbourhood of a, f is the
sum of a power series and of a holomorphic function on C − {a}.
Having said this, let us consider a function f which, instead of being
holomorphic everywhere on a domain G, is so on G − S = G
, where S is
a temporarily finite set. Let ρ a = Res(f, a) be the residue of f at a ∈ S,
i.e. the coefficient of 1/(z − a) in its Laurent series at a. Write g a (z) for the
sum of the terms of degree ≤ −2 – as seen above, it is in fact defined and
holomorphic on C − {a} – and consider the function
g(z) = f (z) −
[g a (z) + ρ a /(z − a)] ;
(5.2)
each g a being holomorphic on C − {a}, g is at least defined on G
. Its only
singular points in G are among the points a ∈ S, but, since like in (2) the
VIII – Cauchy Theory
Hence formula (13) gives a series expansion in the annulus r
< |z| < R
f (z) =
Z
c n z
n
(4.21)
whose coefficients are obtained by integrating f (ζ)ζ
−n−1 either over |ζ| = R
,
or over |ζ| = r
. The choice of the circle is in fact unimportant provided it is
in the interior of the annulus G : r < |z| < R since all circles are obviously
homotopic in G as closed paths. And as the expansion holds for r
< |z| < R
provided that r < r
< R
< R, this means that it holds in all of G. This is
Laurent’s theorem with coefficients given by an integral.
5 – The Residue Formula
(i) The residue formula. A formula (4.10) valid for any closed path μ in G,
any function f holomorphic on G and any z ∈ G, as well as a more general
and extremely classical result, can be proved in a simply connected domain :
Cauchy’s residue formula, an inexhaustible source of exercises and examination questions that, though sometimes subtle, have long become stale. Indeed,
everything can be proved at the same time and G need not be assumed to
be simply connected provided μ is taken to be null-homotopic in G.
First some remarks about the Laurent series of a function f holomorphic
for 0 < |z − a| < R, a being a singular isolated point of f (Chapter VII, § 4,
n
◦ 16). We saw above that it is given by
f (z) =
c n (z − a)
n
with 2πic n =
f (ζ)(ζ − a)
−n−1 dζ
(5.1)
where integration is over any circle t → a + r. exp(2πit) of radius r < R
centered at a. The sum of the terms of degree n < 0, i.e. the polar or singular
part of the Laurent series, is a power series in w = 1/(z − a); it converges
for 0 < |z − a| < R, i.e. for |w| > 1/R; Now, the domain of convergence of
a power series is the interior of a disc; if it converges outside a disc, then it
converges everywhere. Therefore, the series given by the terms of negative
degree in (1) converges for all z = a. So, in the neighbourhood of a, f is the
sum of a power series and of a holomorphic function on C − {a}.
Having said this, let us consider a function f which, instead of being
holomorphic everywhere on a domain G, is so on G − S = G
, where S is
a temporarily finite set. Let ρ a = Res(f, a) be the residue of f at a ∈ S,
i.e. the coefficient of 1/(z − a) in its Laurent series at a. Write g a (z) for the
sum of the terms of degree ≤ −2 – as seen above, it is in fact defined and
holomorphic on C − {a} – and consider the function
g(z) = f (z) −
[g a (z) + ρ a /(z − a)] ;
(5.2)
each g a being holomorphic on C − {a}, g is at least defined on G
. Its only
singular points in G are among the points a ∈ S, but, since like in (2) the