§ 1. Integrals of Holomorphic Functions
25
Hence, if the definition of the γ p , given for 0 < p < n is completed by
setting γ 0 = μ 0 and γ n = μ 1 , we get a sequence of admissible (and even
piecewise linear except for the first and the last one) paths in G
γ 0 = μ 0 , γ 1 , . . . , γ n = μ 1
such that there are linear deformations taking us from each of them to the
next one without leaving G. If the deformation σ we started with is a fixedendpoint homotopy, the intermediate paths γ p clearly also have the same
endpoints as the two given paths. As seen in (iii), a fixed-endpoint linear
homotopy leaves the integral invariant. So the integrals along γ p and γ p+1
are equal for all p. Similarly, if μ 0 and μ 1 are closed and remain so during the
given deformation σ, γ p is closed for all p and remains so during the linear
deformation taking it onto γ p+1 . Hence, once again the integrals are equal.
In conclusion :
Theorem 3. Let G be a domain in C, f a holomorphic function on G and
μ 0 , μ 1 two admissible paths in G. If one of the two following conditions is
satisfied, then the integrals of f along μ 0 and μ 1 are equal :
(a) there is a fixed-endpoint homotopy on G joining μ 0 and μ 1
(b) μ 0 and μ 1 are closed and homotopic in G as closed paths.
Exercise 2 (direct proof of Theorem 3). We keep the above construction
and notation by, for example, supposing that σ is a fixed-endpoint homotopy;
proving this result amounts to showing that integrals along γ p and γ p+1
are equal for all p. For this, take a closed path γ pq made of line segments
connecting a pq , a p,q+1 , a p+1,q+1 , a p+1,q and a pq in the given order; using the
existence of a primitive for f on D pq , show that the integral of f along this
path is zero. Show that the difference between integrals of f along γ p and
γ p+1 is equal to the sum, extended to q, of integrals along γ pq and conclude.
19
Theorem 3 provides a new existence theorem for primitives. For this it
suffices to suppose that condition (b) is satisfied for all μ 0 and μ 1 ; in this
case, any closed path μ is indeed homotopic as a closed path to a “ constant ”
path t → a,where a ∈ G is arbitrarily chosen, so that the integral of f along
μ is zero. Then theorem 1, or its equivalent in terms of closed paths, shows
that f has a primitive on G.
In the next chapter it will be shown that conditions (a) and (b) are equivalent in a more general framework. Domains in which they are satisfied for
all continuous closed paths μ 0 and μ 1 are called simply connected.
Corollary 1. Any holomorphic function on a simply connected domain G
of C has a global primitive on G.
Corollary 2. Let f be a holomorphic function on a simply connected domain G; suppose that f does not vanish on G. Then, there is a holomorphic
19 For similar proofs, see Dieudonn´ e, ´
El´ ements d’analyse, vol. 1, (9.6.3) or Remmert,
Funktionentheorie 2, Chap. 8, § 1, n
◦ 5 and 6.
25
Hence, if the definition of the γ p , given for 0 < p < n is completed by
setting γ 0 = μ 0 and γ n = μ 1 , we get a sequence of admissible (and even
piecewise linear except for the first and the last one) paths in G
γ 0 = μ 0 , γ 1 , . . . , γ n = μ 1
such that there are linear deformations taking us from each of them to the
next one without leaving G. If the deformation σ we started with is a fixedendpoint homotopy, the intermediate paths γ p clearly also have the same
endpoints as the two given paths. As seen in (iii), a fixed-endpoint linear
homotopy leaves the integral invariant. So the integrals along γ p and γ p+1
are equal for all p. Similarly, if μ 0 and μ 1 are closed and remain so during the
given deformation σ, γ p is closed for all p and remains so during the linear
deformation taking it onto γ p+1 . Hence, once again the integrals are equal.
In conclusion :
Theorem 3. Let G be a domain in C, f a holomorphic function on G and
μ 0 , μ 1 two admissible paths in G. If one of the two following conditions is
satisfied, then the integrals of f along μ 0 and μ 1 are equal :
(a) there is a fixed-endpoint homotopy on G joining μ 0 and μ 1
(b) μ 0 and μ 1 are closed and homotopic in G as closed paths.
Exercise 2 (direct proof of Theorem 3). We keep the above construction
and notation by, for example, supposing that σ is a fixed-endpoint homotopy;
proving this result amounts to showing that integrals along γ p and γ p+1
are equal for all p. For this, take a closed path γ pq made of line segments
connecting a pq , a p,q+1 , a p+1,q+1 , a p+1,q and a pq in the given order; using the
existence of a primitive for f on D pq , show that the integral of f along this
path is zero. Show that the difference between integrals of f along γ p and
γ p+1 is equal to the sum, extended to q, of integrals along γ pq and conclude.
19
Theorem 3 provides a new existence theorem for primitives. For this it
suffices to suppose that condition (b) is satisfied for all μ 0 and μ 1 ; in this
case, any closed path μ is indeed homotopic as a closed path to a “ constant ”
path t → a,where a ∈ G is arbitrarily chosen, so that the integral of f along
μ is zero. Then theorem 1, or its equivalent in terms of closed paths, shows
that f has a primitive on G.
In the next chapter it will be shown that conditions (a) and (b) are equivalent in a more general framework. Domains in which they are satisfied for
all continuous closed paths μ 0 and μ 1 are called simply connected.
Corollary 1. Any holomorphic function on a simply connected domain G
of C has a global primitive on G.
Corollary 2. Let f be a holomorphic function on a simply connected domain G; suppose that f does not vanish on G. Then, there is a holomorphic
19 For similar proofs, see Dieudonn´ e, ´
El´ ements d’analyse, vol. 1, (9.6.3) or Remmert,
Funktionentheorie 2, Chap. 8, § 1, n
◦ 5 and 6.