§4. The topology of the functions Arg(z) and Cog z
415
On a surface such as r one can do topology as on the plane: it supports a
distance, namely the standard metric on JR3. We can therefore speak of open
sets, closed sets, continuous functions, convergent sequences, etc. in r, as
well as of continuous maps with values in r; we shall do this in what follows.
One could also consider the graph of the correspondence .cog; this would
be the set of points (z, () of C* x C such that z = exp((). One has to work
in JR4 to represent it geometrically.
(ii) Multiform functions and uniform branches. Instead of speaking of
"correspondences" , mathematicians spoke until recently, and many still speak,
of multiform functions 35 , objects whose first peculiarity is not to be functions;
the adjective "multiform" makes allusion to the fact that these pseudo functions are authorised to take several values at each point, like, for example,
(z2 - 1)1/2 on C and, more generally, the "algebraic functions" obtained by
choosing a polynomial P with complex coefficients in two variables and associating to each x E C the roots y E C of the equation P(x, y) = 0; see the
algebraic curve x 3 - ax 2 + axy - y3 = 0 treated by Newton, who restricted
himself to real variables (Chap. III, nO 14).
Now Newton had already remarked that on the neighbourhood of a point
(xo, Yo) of the curve one can generally find a power series
y = Yo + L an(x - xo)n
which satisfies the given equation P(x, y) = 0 identically. Although the equation P(x, y) = 0 does not allow one to consider y as a true function of x,
there nevertheless exist excellent functions y = f(x) which satisfy it. In the
simplest case of the correspondence x 2 + y2 = 1 between points of JR, a correspondence whose graph is the circle with centre 0 and radius 1, the formulae
y = (1 - x 2 )1/2 and y = -(1 - X 2 )1/2 define two such functions on [-1,1].
One is thus led to define a uniform branch of a multiform function or
correspondence: any true function, with only one value, whose graph is contained in that of the given correspondence. In the case of the argument, a
uniform branch on a set G c C* is thus a function A : G ~ JR satisfying
(21.4)
z = Izl exp[i.A(z)]
for all z E G;
35 Instead of considering Cog z as a (false) function defined on C*, one can save
the situation by considering it as a true function defined on its own graph,
namely (z, () ........ (, since, on the latter, we have exp(() = z. In the theory of
analytic functions this artifice is the origin of the invention of Riemann surfaces of
algebraic functions. Instead of considering, for example, (z4 _1)1/7 as a function
of z E C, which in the strict sense it is not, one works on the surface G in C 2
defined by the equation C = Z4 - 1 and studies the function (z, () ........ ( thereon,
and, more generally, the functions (z, () ........ f(z, () where f is a rational function
of two variables. The true Riemann surface is a little more complicated, but is all
the same the initial motivation for the construction and the theory of algebraic
functions of a complex variable. See Chap. XI in Vol. III.
415
On a surface such as r one can do topology as on the plane: it supports a
distance, namely the standard metric on JR3. We can therefore speak of open
sets, closed sets, continuous functions, convergent sequences, etc. in r, as
well as of continuous maps with values in r; we shall do this in what follows.
One could also consider the graph of the correspondence .cog; this would
be the set of points (z, () of C* x C such that z = exp((). One has to work
in JR4 to represent it geometrically.
(ii) Multiform functions and uniform branches. Instead of speaking of
"correspondences" , mathematicians spoke until recently, and many still speak,
of multiform functions 35 , objects whose first peculiarity is not to be functions;
the adjective "multiform" makes allusion to the fact that these pseudo functions are authorised to take several values at each point, like, for example,
(z2 - 1)1/2 on C and, more generally, the "algebraic functions" obtained by
choosing a polynomial P with complex coefficients in two variables and associating to each x E C the roots y E C of the equation P(x, y) = 0; see the
algebraic curve x 3 - ax 2 + axy - y3 = 0 treated by Newton, who restricted
himself to real variables (Chap. III, nO 14).
Now Newton had already remarked that on the neighbourhood of a point
(xo, Yo) of the curve one can generally find a power series
y = Yo + L an(x - xo)n
which satisfies the given equation P(x, y) = 0 identically. Although the equation P(x, y) = 0 does not allow one to consider y as a true function of x,
there nevertheless exist excellent functions y = f(x) which satisfy it. In the
simplest case of the correspondence x 2 + y2 = 1 between points of JR, a correspondence whose graph is the circle with centre 0 and radius 1, the formulae
y = (1 - x 2 )1/2 and y = -(1 - X 2 )1/2 define two such functions on [-1,1].
One is thus led to define a uniform branch of a multiform function or
correspondence: any true function, with only one value, whose graph is contained in that of the given correspondence. In the case of the argument, a
uniform branch on a set G c C* is thus a function A : G ~ JR satisfying
(21.4)
z = Izl exp[i.A(z)]
for all z E G;
35 Instead of considering Cog z as a (false) function defined on C*, one can save
the situation by considering it as a true function defined on its own graph,
namely (z, () ........ (, since, on the latter, we have exp(() = z. In the theory of
analytic functions this artifice is the origin of the invention of Riemann surfaces of
algebraic functions. Instead of considering, for example, (z4 _1)1/7 as a function
of z E C, which in the strict sense it is not, one works on the surface G in C 2
defined by the equation C = Z4 - 1 and studies the function (z, () ........ ( thereon,
and, more generally, the functions (z, () ........ f(z, () where f is a rational function
of two variables. The true Riemann surface is a little more complicated, but is all
the same the initial motivation for the construction and the theory of algebraic
functions of a complex variable. See Chap. XI in Vol. III.