46
VIII – Cauchy Theory
This type of argument, heavily exploited by Riemann in his theory of
algebraic functions – in his days, he did not have any choice – and by his
many successors, has, nonetheless, supplied several useful formulas as we will
have occasion to see later.
(ii) Cauchy’s integral formula : the general case. To generalize Cauchy’s
integral formula (4.10) with respect to the circle, we apply theorem 5 to the
function g(ζ) = f (ζ)/(ζ − w), where w ∈ G − S is given. It is holomorphic on
G−{w} ∪ S and it all amounts to calculating the residues. First, Res(g, w) =
f (w) clearly holds since f (ζ) = f (w)+f
(w)(ζ −w)+. . . in the neighbourhood
of w (Taylor series). In the neighbourhood of a point a ∈ S,
1/(ζ − w) = −1/ [(w − a) − (ζ − a)] = −
m ∈ N
(w − a)
−m−1 (ζ − a)
m
if |ζ − a| < |w − a|; Hence, if
f (ζ) =
n ∈ Z
c n (ζ − a)
n
(5.6)
in the neighbourhood of the singular point a, then the associativity theorem
for absolutely convergent series (Chap. II, n
◦ 18) shows that
g(ζ) = −
n ∈ Z, m ≥ 0
c n (w − a)
−m−1 (ζ − a)
m+n .
The residue of g is obtained by grouping together the terms for which m+n =
−1 (same reference), and so
Res(g, a) = −
m ∈ N
c −m−1 (w − a)
−m−1 = −
n<0
c n (w − a)
n .
(5.7)
This is the value up to sign at w of the polar part
31 of the Laurent series of f
with respect to a.
This formula simplifies if f has a simple pole at a, i.e. if its Laurent
series with respect to a reduces to its term of degree −1 , leaving Res(g, a) =
−c −1 /(w − a) = − Res(f, a)/(w − a). Hence formula (5) applied to g gives
the following result :
Theorem 6. Let G be a domain C, S a closed and discrete subset of G and
f a holomorphic function on G − S with simple poles at every point of S. If
w ∈ G − S and if μ is a closed path in G − S ∪ {w} null-homotopic in G, then
1
2πi
μ
f (ζ)
ζ − w
dζ = Ind μ (w)f (w) +
a∈S
Ind μ (a)
Res(f, a)
a − w
.
(5.8)
31 It was shown above that this polar part converges in C − {a}.