§ 1. Integrals of Holomorphic Functions
9
Conversely, if F is defined on D using this formula, then F is holomorphic
and satisfies F
= f . To see this, observe that the function of (t, x, y) under
the sign
is C
1 . This allows us to differentiate under the sign
with respect
to x or y; setting D 1 = d/dx and D = d/dt, omitting the limits of integration
and noting that D 1 z = 1, we get
D 1 F (z) =
D 1 [f (tx, ty)z] dt =
[D 1 f (tz).tz + f (tz)] dt ;
since f is holomorphic, D 1 f = f
and f
(tz)z is the derivative of f (tz) with
respect to t; hence, integrating by parts, it follows that
D 1 F (z) =
f (tz)dt +
t.D [f (tz)] dt =
f (tz)dt + tf (tz)
1
0
−
f (tz)dt ,
and so D 1 F = f . If D 1 is now replaced by D 2 = d/dy, the calculation
remains the same except that D 2 f = if
. Then D 2 F = if ; as a result, F
satisfies Cauchy’s condition and F
(z) = D 1 F (z) = f (z), q.e.d.
This method applies more generally to any star G, i.e. which does not
have any point a such that, for all z ∈ G, the line segment [a, z] is contained
in G : replace tz by a + t(z − a) in (2). It is for example the case of an open
convex subset of C − R + by choosing a on the negative real axis, etc. We will,
however, find further down a less restrictive result regarding G.
Let us return to the general case. The results obtained above mean that
every holomorphic function f on an open set G has a primitive in the neighbourhood of each point of G; but, as already seen (Chapter IV, § 4) for 1/z
and its pseudo-primitive Log z, this local result in no way implies the existence of a global primitive, i.e. valid on all of G; we will return to this point
later.
(ii) Integration along a path. Admissible paths. Let f be a function defined
and holomorphic on an open connected subset G of C, i.e. a domain, and
suppose that f has a primitive F in G. If we were on R, the FT
F (z) − F (a) =
z
a
f (ζ)dζ ⇐⇒ F
(z) = f (z)
(2.3)
where, despite the notation, z and the integration variable ζ are reals, would
allow us to calculate F up to an additive constant. But, at first sight, integrating from a point a ∈ G to another point z ∈ G is not well-defined on
C.
9
Conversely, if F is defined on D using this formula, then F is holomorphic
and satisfies F
= f . To see this, observe that the function of (t, x, y) under
the sign
is C
1 . This allows us to differentiate under the sign
with respect
to x or y; setting D 1 = d/dx and D = d/dt, omitting the limits of integration
and noting that D 1 z = 1, we get
D 1 F (z) =
D 1 [f (tx, ty)z] dt =
[D 1 f (tz).tz + f (tz)] dt ;
since f is holomorphic, D 1 f = f
and f
(tz)z is the derivative of f (tz) with
respect to t; hence, integrating by parts, it follows that
D 1 F (z) =
f (tz)dt +
t.D [f (tz)] dt =
f (tz)dt + tf (tz)
1
0
−
f (tz)dt ,
and so D 1 F = f . If D 1 is now replaced by D 2 = d/dy, the calculation
remains the same except that D 2 f = if
. Then D 2 F = if ; as a result, F
satisfies Cauchy’s condition and F
(z) = D 1 F (z) = f (z), q.e.d.
This method applies more generally to any star G, i.e. which does not
have any point a such that, for all z ∈ G, the line segment [a, z] is contained
in G : replace tz by a + t(z − a) in (2). It is for example the case of an open
convex subset of C − R + by choosing a on the negative real axis, etc. We will,
however, find further down a less restrictive result regarding G.
Let us return to the general case. The results obtained above mean that
every holomorphic function f on an open set G has a primitive in the neighbourhood of each point of G; but, as already seen (Chapter IV, § 4) for 1/z
and its pseudo-primitive Log z, this local result in no way implies the existence of a global primitive, i.e. valid on all of G; we will return to this point
later.
(ii) Integration along a path. Admissible paths. Let f be a function defined
and holomorphic on an open connected subset G of C, i.e. a domain, and
suppose that f has a primitive F in G. If we were on R, the FT
F (z) − F (a) =
z
a
f (ζ)dζ ⇐⇒ F
(z) = f (z)
(2.3)
where, despite the notation, z and the integration variable ζ are reals, would
allow us to calculate F up to an additive constant. But, at first sight, integrating from a point a ∈ G to another point z ∈ G is not well-defined on
C.