§ 2. Cauchy’s Integral Formulas
47
This type of argument has several versions.
Theorem 6 bis. Let G be a simply connected domain, f a holomorphic
function on G and μ a closed admissible path in G. Then
1
2πi
μ
f (ζ)
(ζ − w) n+1 dζ = Ind μ (w).f
(n) (w)/n!
(5.9)
for all w ∈ G − Supp(μ) and for all n ∈ N.
This result is the residue theorem applied to the function f (ζ)/(ζ −w)
n+1 ;
indeed, it has at most one singular point in G : a pole at z = w, with residue
f
(n) (w)/n!, since Taylor’s formula
f (z)/(z − w)
n+1 = (z − w)
−n−1
(z − w)
p f
(p) (w)/p!
shows that the coefficient of 1/(z − w) is equal to f
(n) (w)/n!.
(iii) The number of zeros and poles of a function. The residue formula has
several other immediate consequences. Consider for example a meromorphic
function f on a domain G; f being holomorphic and without any zeros on
G − S, this means that there exists a discrete and closed subset S of G such
that the points of S are zeros or poles, but not essential singularities, of f . The
function f
(z)/f (z) is then holomorphic on G − S and its only singularities
are the points a ∈ S. In the neighbourhood of any point a ∈ G, there is a
series expansion
f (z) = c p (z − a)
p + c p+1 (z − a)
p+1 + . . .
(5.10)
with c p = 0; the integer p is ≥ 0 if f is holomorphic at a; it is > 0 if f (a) = 0,
and it is < 0 if a is a pole of f . Denote it by v a (f ), so that if S is finite, then
f (z) = u(z)
(z − a)
va(f ) ,
where the function u is holomorphic everywhere and = 0 everywhere on G,
and so has an inverse in the ring of meromorphic functions on G. This is
the analogue of the decomposition of an integer into primes.
32 Having said
this, let us apply the residue formula to the function f
(z)/f (z), holomorphic
outside S. By (10),
f (z) = (z − a)
p g(z) ,
where g is holomorphic and non-zero at a, and so
32 for extension to meromorphic functions in divisibility theory , see Remmert,
Funktionentheorie 2, Chap. 3 and 4, where it is in particular shown that every meromorphic function on a domain G is the quotients of two holomorphic
functions on G having no common zeros.
47
This type of argument has several versions.
Theorem 6 bis. Let G be a simply connected domain, f a holomorphic
function on G and μ a closed admissible path in G. Then
1
2πi
μ
f (ζ)
(ζ − w) n+1 dζ = Ind μ (w).f
(n) (w)/n!
(5.9)
for all w ∈ G − Supp(μ) and for all n ∈ N.
This result is the residue theorem applied to the function f (ζ)/(ζ −w)
n+1 ;
indeed, it has at most one singular point in G : a pole at z = w, with residue
f
(n) (w)/n!, since Taylor’s formula
f (z)/(z − w)
n+1 = (z − w)
−n−1
(z − w)
p f
(p) (w)/p!
shows that the coefficient of 1/(z − w) is equal to f
(n) (w)/n!.
(iii) The number of zeros and poles of a function. The residue formula has
several other immediate consequences. Consider for example a meromorphic
function f on a domain G; f being holomorphic and without any zeros on
G − S, this means that there exists a discrete and closed subset S of G such
that the points of S are zeros or poles, but not essential singularities, of f . The
function f
(z)/f (z) is then holomorphic on G − S and its only singularities
are the points a ∈ S. In the neighbourhood of any point a ∈ G, there is a
series expansion
f (z) = c p (z − a)
p + c p+1 (z − a)
p+1 + . . .
(5.10)
with c p = 0; the integer p is ≥ 0 if f is holomorphic at a; it is > 0 if f (a) = 0,
and it is < 0 if a is a pole of f . Denote it by v a (f ), so that if S is finite, then
f (z) = u(z)
(z − a)
va(f ) ,
where the function u is holomorphic everywhere and = 0 everywhere on G,
and so has an inverse in the ring of meromorphic functions on G. This is
the analogue of the decomposition of an integer into primes.
32 Having said
this, let us apply the residue formula to the function f
(z)/f (z), holomorphic
outside S. By (10),
f (z) = (z − a)
p g(z) ,
where g is holomorphic and non-zero at a, and so
32 for extension to meromorphic functions in divisibility theory , see Remmert,
Funktionentheorie 2, Chap. 3 and 4, where it is in particular shown that every meromorphic function on a domain G is the quotients of two holomorphic
functions on G having no common zeros.