378
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
and so
(14.20)
.cog z = log Izl + i. arg z
where log Izl is the usual logarithm on lR:+-. The ambiguity in the definition
of arg z causes exactly the same problem for the complex .cog. We will find
it again a propos the primitive of the function 1/(x - a) for a E C, and
this is how Johann Bernoulli introduced it in 1702 (Cantor, p. 362, a propos
arctan). Calculations with complex logarithms or, what comes to the same,
with arguments of complex numbers, remains, three centuries later, one of
the most notorious sources of errors.
We saw in nO 11, Theorem 8 bis, that, as in the real case, the series
u = z - z2/2 + z3/3 - ... satisfies exp u = 1 + z for z E C, Izl < 1. We deduce
that
(14.21 )
.cog(1 + z)
2ki7r + z - z2/2 + z3/3 - ...
for z E C, Izl < 1.
It is tempting to write Newton's binomial formula in the form
(14.22)
(1 + z)S = exp[s . .cog(1 + z)]
for z and s complex. Here again, handle with caution. If indeed one defines
(14.23)
as = exp(s . .cog a)
for a and s complex, a =I- 0, the ambiguity inherent in the definition of the
complex log carries over to the definition of as, which will be defined only up
to a factor of the form exp(2k7ris) = exp(27ris)k. If s is a rational number,
one obtains a finite number of possible values, but in the general case the
map k f-+ exp(2k7ris)k of Z into C is injective, whence a countable infinity of
"determinations" of as. Here again, a notorious source of errors.
Formula (21), which lets one calculate .cog on a neighbourhood of 1,
extends to all other points a E C*: every z sufficiently close to a is of the
form z = a(1 + u) with lui < 1, whence 25
(14.21 ')
.cog z = .cog a - L(1 - z/a)n/n,
a power series converging for Iz - al < lal. There are therefore, on a neighbourhood of a, analytic functions whose values at every point z form a subset
of the set of possible values of .cog z. Every analytic function which possesses
this property on an open subset U c C* is called an analytic or uniform
branch of .cog on U. The existence of such functions imposes serious restrictions on the "topology" of U, as we shall see in § 4.
25 This means that all the possible values of Cog z are obtained by adding to the
series any value of Cog a.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
and so
(14.20)
.cog z = log Izl + i. arg z
where log Izl is the usual logarithm on lR:+-. The ambiguity in the definition
of arg z causes exactly the same problem for the complex .cog. We will find
it again a propos the primitive of the function 1/(x - a) for a E C, and
this is how Johann Bernoulli introduced it in 1702 (Cantor, p. 362, a propos
arctan). Calculations with complex logarithms or, what comes to the same,
with arguments of complex numbers, remains, three centuries later, one of
the most notorious sources of errors.
We saw in nO 11, Theorem 8 bis, that, as in the real case, the series
u = z - z2/2 + z3/3 - ... satisfies exp u = 1 + z for z E C, Izl < 1. We deduce
that
(14.21 )
.cog(1 + z)
2ki7r + z - z2/2 + z3/3 - ...
for z E C, Izl < 1.
It is tempting to write Newton's binomial formula in the form
(14.22)
(1 + z)S = exp[s . .cog(1 + z)]
for z and s complex. Here again, handle with caution. If indeed one defines
(14.23)
as = exp(s . .cog a)
for a and s complex, a =I- 0, the ambiguity inherent in the definition of the
complex log carries over to the definition of as, which will be defined only up
to a factor of the form exp(2k7ris) = exp(27ris)k. If s is a rational number,
one obtains a finite number of possible values, but in the general case the
map k f-+ exp(2k7ris)k of Z into C is injective, whence a countable infinity of
"determinations" of as. Here again, a notorious source of errors.
Formula (21), which lets one calculate .cog on a neighbourhood of 1,
extends to all other points a E C*: every z sufficiently close to a is of the
form z = a(1 + u) with lui < 1, whence 25
(14.21 ')
.cog z = .cog a - L(1 - z/a)n/n,
a power series converging for Iz - al < lal. There are therefore, on a neighbourhood of a, analytic functions whose values at every point z form a subset
of the set of possible values of .cog z. Every analytic function which possesses
this property on an open subset U c C* is called an analytic or uniform
branch of .cog on U. The existence of such functions imposes serious restrictions on the "topology" of U, as we shall see in § 4.
25 This means that all the possible values of Cog z are obtained by adding to the
series any value of Cog a.