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IV - Powers, Exponentials, Logarithms, Trigonometric Functions
§4. The topology of the functions Arg(z) and Cog z
21 - In nO 14 we defined the argument of a complex number z -# 0 as being
any real number t = arg(z) such that
(21.1)
z = Izl exp(it)
and showed that the problem admitted infinitely many solutions, differing
one from the other by arbitrary multiples of 27r. The standard notation arg
does not denote a true function; it is a correspondence, in the sense of Chap. I,
between elements of the set C* and elements of 1R; for any z E C* , the symbol
arg(z) should therefore denote the set of t E IR such that z = Izl exp(it) even
if, in practice, it almost always denotes one of these possible values. To avoid
confusion, we shall write Arg(z) for this set, so that
(21.2)
t E Arg(z) {==} z = Izl exp(it).
Similarly we agree that the notation .cog z denotes the set of possible values
of the complex log:
(21.2')
( E .cog z {==} z = exp().
Everyone who has taught the theory of holomorphic functions knows that
arguments and, what comes to the same, logarithms of complex numbers, are
one of the most frequent sources of theoretical incomprehension and of errors in calculation. We may therefore be pardoned for developing the subject
quite untraditionally. This will also be an occasion to familiarise the reader
with some topological techniques of much more general relevance.
(i) Graph of z ~ Arg(z). A correspondence is defined by exhibiting
its graph, namely, in this case, the set of pairs (z, t) E C* x lR such that
z = Izl exp(it). The Cartesian product can be represented in the usual threedimensional space by choosing a system of rectangular coordinates (x, y, t),
identifying C* with the horizontal plane Oxy with the origin deleted, and lR
with the vertical t axis; the product is then 1R 3 less the vertical axis. The
graph r of the correspondence Arg is then the set of points (x, y, t) E lR 3
such that
(21.3)
x = r cos t, y = r sin t
where r = (x 2 + y2)1/2 -# 0;
the vertical through a point (x, y) meets it in infinitely many points, each
distant one from the other by a multiple of 27r. This is a helicoidal surface
analogous to an inclined ramp of infinite size rotating at constant angular
speed about the t axis and prolonged to infinity above and below; one could
probably feature it in a science-fiction novel. The intersection of r with the
surface of a circular cylinder with axis Ot is a helix of pitch 27r winding
indefinitely around the cylinder.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
§4. The topology of the functions Arg(z) and Cog z
21 - In nO 14 we defined the argument of a complex number z -# 0 as being
any real number t = arg(z) such that
(21.1)
z = Izl exp(it)
and showed that the problem admitted infinitely many solutions, differing
one from the other by arbitrary multiples of 27r. The standard notation arg
does not denote a true function; it is a correspondence, in the sense of Chap. I,
between elements of the set C* and elements of 1R; for any z E C* , the symbol
arg(z) should therefore denote the set of t E IR such that z = Izl exp(it) even
if, in practice, it almost always denotes one of these possible values. To avoid
confusion, we shall write Arg(z) for this set, so that
(21.2)
t E Arg(z) {==} z = Izl exp(it).
Similarly we agree that the notation .cog z denotes the set of possible values
of the complex log:
(21.2')
( E .cog z {==} z = exp().
Everyone who has taught the theory of holomorphic functions knows that
arguments and, what comes to the same, logarithms of complex numbers, are
one of the most frequent sources of theoretical incomprehension and of errors in calculation. We may therefore be pardoned for developing the subject
quite untraditionally. This will also be an occasion to familiarise the reader
with some topological techniques of much more general relevance.
(i) Graph of z ~ Arg(z). A correspondence is defined by exhibiting
its graph, namely, in this case, the set of pairs (z, t) E C* x lR such that
z = Izl exp(it). The Cartesian product can be represented in the usual threedimensional space by choosing a system of rectangular coordinates (x, y, t),
identifying C* with the horizontal plane Oxy with the origin deleted, and lR
with the vertical t axis; the product is then 1R 3 less the vertical axis. The
graph r of the correspondence Arg is then the set of points (x, y, t) E lR 3
such that
(21.3)
x = r cos t, y = r sin t
where r = (x 2 + y2)1/2 -# 0;
the vertical through a point (x, y) meets it in infinitely many points, each
distant one from the other by a multiple of 27r. This is a helicoidal surface
analogous to an inclined ramp of infinite size rotating at constant angular
speed about the t axis and prolonged to infinity above and below; one could
probably feature it in a science-fiction novel. The intersection of r with the
surface of a circular cylinder with axis Ot is a helix of pitch 27r winding
indefinitely around the cylinder.