8
VIII – Cauchy Theory
Indeed, by (7), formula (12) is exact if the derivatives f
, g
and p
are
interpreted as linear maps from R
2 to R
2 and the right hand side as the
composition of functions f
[g(z)] and g
(z). By assumption, these maps are
C-linear. But the composition of two C-linear maps h → ah and h → bh on C
gives a map h → abh; formula (12) can, therefore, be obtained by substituting
the corresponding complex numbers to the maps p
(z), etc..
As shown in Chap. VII, saying that a function f is holomorphic on an
open set G amounts to saying that it is analytic on G, i.e. that, for each a ∈ G,
it has a power series expansion f (z) =
n≥0 c n (z − a)
n which converges and
represents it on a disc centered at a and in fact on the largest disc centered
at a contained in G; the power series which represents f in a neighbourhood
of a is just its Taylor series
f (z) =
n≥0
f
(n) (a)(z − a)
[n] ,
where, we remind the reader that z
[n] = z
n /n!. The terms “ holomorphic ”
and “ analytic ” are, therefore, synonymous.
Nonetheless, all the results that we will prove in this chapter are based
only on the initial definition of holomorphic functions, in other words, do
not use their analyticity, a result that we will recover by Cauchy’s traditional method. Hence, we will maintain the strict distinction between
“ holomorphic ” functions and “ analytic ” functions until we again prove the
equivalence of these two notions.
2 – The Problem of Primitives
(i) Local primitives of a holomorphic function. One of the basic problems in
the theory of holomorphic functions is to find a function f holomorphic on
an open subset U of C a primitive of f on U , i.e. a holomorphic function
F such that F
= f . If U is a disc centered at a, the problem always has a
solution since a power series can be differentiated term by term (Chap. II,
n
◦ 19) :
f (z) =
n≥0
c n (z − a)
n
⇐⇒ F (z) = c +
n≥0
c n (z − a)
n+1 /(n + 1) ,
(2.1)
where c is an arbitrary constant. A proof which does not use analyticity
and which generalizes to differential forms consists in observing that if f
is holomorphic on the disc D : |z| < R and if F
= f , then, by (11), the
derivative of the function t → F (tz), defined at least on [0, 1] for a given
z ∈ D, is F
(tz)z = f (tz)z; The FT then shows that, when F (0) = 0,
F (z) =
1
0
f (tz)zdt
(2.2)
for all z ∈ D.
VIII – Cauchy Theory
Indeed, by (7), formula (12) is exact if the derivatives f
, g
and p
are
interpreted as linear maps from R
2 to R
2 and the right hand side as the
composition of functions f
[g(z)] and g
(z). By assumption, these maps are
C-linear. But the composition of two C-linear maps h → ah and h → bh on C
gives a map h → abh; formula (12) can, therefore, be obtained by substituting
the corresponding complex numbers to the maps p
(z), etc..
As shown in Chap. VII, saying that a function f is holomorphic on an
open set G amounts to saying that it is analytic on G, i.e. that, for each a ∈ G,
it has a power series expansion f (z) =
n≥0 c n (z − a)
n which converges and
represents it on a disc centered at a and in fact on the largest disc centered
at a contained in G; the power series which represents f in a neighbourhood
of a is just its Taylor series
f (z) =
n≥0
f
(n) (a)(z − a)
[n] ,
where, we remind the reader that z
[n] = z
n /n!. The terms “ holomorphic ”
and “ analytic ” are, therefore, synonymous.
Nonetheless, all the results that we will prove in this chapter are based
only on the initial definition of holomorphic functions, in other words, do
not use their analyticity, a result that we will recover by Cauchy’s traditional method. Hence, we will maintain the strict distinction between
“ holomorphic ” functions and “ analytic ” functions until we again prove the
equivalence of these two notions.
2 – The Problem of Primitives
(i) Local primitives of a holomorphic function. One of the basic problems in
the theory of holomorphic functions is to find a function f holomorphic on
an open subset U of C a primitive of f on U , i.e. a holomorphic function
F such that F
= f . If U is a disc centered at a, the problem always has a
solution since a power series can be differentiated term by term (Chap. II,
n
◦ 19) :
f (z) =
n≥0
c n (z − a)
n
⇐⇒ F (z) = c +
n≥0
c n (z − a)
n+1 /(n + 1) ,
(2.1)
where c is an arbitrary constant. A proof which does not use analyticity
and which generalizes to differential forms consists in observing that if f
is holomorphic on the disc D : |z| < R and if F
= f , then, by (11), the
derivative of the function t → F (tz), defined at least on [0, 1] for a given
z ∈ D, is F
(tz)z = f (tz)z; The FT then shows that, when F (0) = 0,
F (z) =
1
0
f (tz)zdt
(2.2)
for all z ∈ D.