§ 2. Cauchy’s Integral Formulas
55
at a itself. In particular, any rational function f (z) can be interpreted on ˆ
C
as a function whose only singularities are its poles, of which are there are
necessarily finitely many since ˆ
C is compact; and we have seen that they are
characterized by this property : rational functions are identical to meromorphic function on ˆ
C.
To give the reader a less trivial example connected to the theory of elliptic
functions, we choose a lattice L in C (Chapter II, § 3, n
◦ 23) and we consider
the set C/L of equivalence classes mod L, obtained by regarding identical two
numbers z
, z
∈ C such that z
− z
∈ L. Writing p for the map C −→ C/L
associating to each z ∈ C its class mod L, a topology can be defined on C/L
by setting a subset U ⊂ C/L to be open if and only if p
−1 (U ) is open in C :
to ensure the continuity of p, we confine ourselves to the minimum required.
The space C/L is compact
36 since, if we choose a compact subset K in C
meeting all the classes mod L (for example a closed parallelogram generated
by two basis vectors
37 for L), then p(K) = C/L. As p is continuous, since BW
or BL holds for K it holds for C/L (see Theorem 11 of Chapter III, § 3, n
◦ 9,
whose proof generalizes immediately). Having said this, a function f defined
on such an open set is, by definition, holomorphic if and only if the function
z → f [p(z)], defined on the open subset p
−1 (U ) of C, is holomorphic in the
usual sense (and doubly periodic since it is constant on the classes mod L). In
this case, it would be easy to explain what is meant by the pole of a function
defined in the neighbourhood of a point of C/L, but not at the point itself;
transposing what we are doing to C would be sufficient. In particular, a
meromorphic function f on C/L only has finitely many poles since C/L is
compact; composing it with p, we get a doubly periodic and meromorphic
function on C : as we will see in Chap. XII, this is precisely what is called
an elliptic function of the lattice L and we will show that, in this case, the
meromorphic functions on C/L are just the rational functions in ℘ L (z) and
℘
L (z), where ℘ L is the Weierstrass function of L (Chapter II, § 3, n
◦ 23).
Hence all this is only a matter of translation, but this point of view has
proved itself exceedingly fruitful in much more general cases in most of which
the construction is not at all as simple as in the last two examples, starting
with the case of algebraic functions of one variable studied by Riemann, i.e.
36 It is also necessary to show that C/L satisfies the Hausdorff axiom : that two
distinct points have disjoint neighbourhoods; this is equivalent to saying that if
a, a
∈ C do not belong to the same class mod L, then there are discs D and D
with centered respectively at a and a
such that (D + L) ∩ (D
+ L) = ∅, which
is immediate. I have not mention this axiom in the Appendix to Chap. III in
order not to steer the reader towards situations rarely encountered in everyday
mathematics, but it is nonetheless fundamental. It obviously holds in all metric
spaces.
37 Topologically, C/L can, therefore, be obtained by taking a period-parallelogram
P and by “ gluing ” its parallel sides pairwise; by gluing two parallel sides, we
get a cylinder and by gluing the end circles, a ring. In other words, C/L is
homeomorphic to the surface of a torus in R
3 .
55
at a itself. In particular, any rational function f (z) can be interpreted on ˆ
C
as a function whose only singularities are its poles, of which are there are
necessarily finitely many since ˆ
C is compact; and we have seen that they are
characterized by this property : rational functions are identical to meromorphic function on ˆ
C.
To give the reader a less trivial example connected to the theory of elliptic
functions, we choose a lattice L in C (Chapter II, § 3, n
◦ 23) and we consider
the set C/L of equivalence classes mod L, obtained by regarding identical two
numbers z
, z
∈ C such that z
− z
∈ L. Writing p for the map C −→ C/L
associating to each z ∈ C its class mod L, a topology can be defined on C/L
by setting a subset U ⊂ C/L to be open if and only if p
−1 (U ) is open in C :
to ensure the continuity of p, we confine ourselves to the minimum required.
The space C/L is compact
36 since, if we choose a compact subset K in C
meeting all the classes mod L (for example a closed parallelogram generated
by two basis vectors
37 for L), then p(K) = C/L. As p is continuous, since BW
or BL holds for K it holds for C/L (see Theorem 11 of Chapter III, § 3, n
◦ 9,
whose proof generalizes immediately). Having said this, a function f defined
on such an open set is, by definition, holomorphic if and only if the function
z → f [p(z)], defined on the open subset p
−1 (U ) of C, is holomorphic in the
usual sense (and doubly periodic since it is constant on the classes mod L). In
this case, it would be easy to explain what is meant by the pole of a function
defined in the neighbourhood of a point of C/L, but not at the point itself;
transposing what we are doing to C would be sufficient. In particular, a
meromorphic function f on C/L only has finitely many poles since C/L is
compact; composing it with p, we get a doubly periodic and meromorphic
function on C : as we will see in Chap. XII, this is precisely what is called
an elliptic function of the lattice L and we will show that, in this case, the
meromorphic functions on C/L are just the rational functions in ℘ L (z) and
℘
L (z), where ℘ L is the Weierstrass function of L (Chapter II, § 3, n
◦ 23).
Hence all this is only a matter of translation, but this point of view has
proved itself exceedingly fruitful in much more general cases in most of which
the construction is not at all as simple as in the last two examples, starting
with the case of algebraic functions of one variable studied by Riemann, i.e.
36 It is also necessary to show that C/L satisfies the Hausdorff axiom : that two
distinct points have disjoint neighbourhoods; this is equivalent to saying that if
a, a
∈ C do not belong to the same class mod L, then there are discs D and D
with centered respectively at a and a
such that (D + L) ∩ (D
+ L) = ∅, which
is immediate. I have not mention this axiom in the Appendix to Chap. III in
order not to steer the reader towards situations rarely encountered in everyday
mathematics, but it is nonetheless fundamental. It obviously holds in all metric
spaces.
37 Topologically, C/L can, therefore, be obtained by taking a period-parallelogram
P and by “ gluing ” its parallel sides pairwise; by gluing two parallel sides, we
get a cylinder and by gluing the end circles, a ring. In other words, C/L is
homeomorphic to the surface of a torus in R
3 .