418
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
on the same open set G. We shall show that A(z) and then L(z) are continuous functions of z on G, which will establish the result. Here again this
is "geometrically obvious", but a rigorous, i.e. analytic, proof, rests on a
property of the complex logarithm which no sketch can provide.
Let us work on a neighbourhood of an arbitrary a E G; we have shown
above [nO 14, section (x)] that, on the disc D(a) : Iz - al < lal, the values of
.cog z are given by
(21.8)
.cog z = .cog a - 2:(1- z/a)n In.
If we choose .cog a = L(a) in (8), where L(a) is given by (7), the right hand
side is a continuous function of z on D(a), so its imaginary part is too; the
latter being equal to A(a) for z = a and IA(a)1 being < 7r, the imaginary
part of the right hand side of (8) is again, in absolute value, < 7r on a
neighbourhood of a: it is thus A(z), whence it follows that
(21.9)
L(z) = L(a) - 2:(1- z/a)n /n
for Iz - al sufficiently small. Hence the continuity of the function (7) and thus
of A(z) on the open set considered. This argument even shows much more:
every uniform branch of .cog is analytic.
(vi) This result shows a fortiori that there exist uniform branches of the
argument on every open disc D C C*: remove from C a half-line not meeting
D, for example llL if D is contained in the half plane Re(z) > 0, a case to
which one can always reduce by a rotation around the origin. In this particular case, it is clear that, for zED, one can choose A(z) in the interval
] - 7r /2, 7r /2[. In the general case, one sees that if A(z) is a uniform branch
of the argument in D, then the set A(D) of values taken by A(z) for zED
is an interval of length < 7r.
(vii) Liftings of a path. By definition, a continuous path in C is a continuous map of a compact interval I = [a, b] of lR into C; this definition
extends trivially to any metric space, and in particular to the graph r of the
correspondence Arg introduced above. If t 1--+ 'Y(t) is a path in r, we have
(21.10)
'Y(t) = ("to(t),A(t»
where 'Yo(t) E C* and A(t) E lR are continuous functions of t; it is clear that
'Yo : I - + C* is a continuous path in C* and that, for all tEl, one must
have A(t) E Arg['Yo(t)], the set of possible arguments of 'Yo(t).
If, conversely, one has a continuous path ' Y in C* and chooses a number
A(t) E Arg!"f(t)] for all tEl so that the function A(t) is continuous, i.e. is a
uniform branch of the argument along'Y (similar definition to the case of .cog),
then t 1--+ ("t(t), A(t» is a continuous path in r which projects horizontally
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
on the same open set G. We shall show that A(z) and then L(z) are continuous functions of z on G, which will establish the result. Here again this
is "geometrically obvious", but a rigorous, i.e. analytic, proof, rests on a
property of the complex logarithm which no sketch can provide.
Let us work on a neighbourhood of an arbitrary a E G; we have shown
above [nO 14, section (x)] that, on the disc D(a) : Iz - al < lal, the values of
.cog z are given by
(21.8)
.cog z = .cog a - 2:(1- z/a)n In.
If we choose .cog a = L(a) in (8), where L(a) is given by (7), the right hand
side is a continuous function of z on D(a), so its imaginary part is too; the
latter being equal to A(a) for z = a and IA(a)1 being < 7r, the imaginary
part of the right hand side of (8) is again, in absolute value, < 7r on a
neighbourhood of a: it is thus A(z), whence it follows that
(21.9)
L(z) = L(a) - 2:(1- z/a)n /n
for Iz - al sufficiently small. Hence the continuity of the function (7) and thus
of A(z) on the open set considered. This argument even shows much more:
every uniform branch of .cog is analytic.
(vi) This result shows a fortiori that there exist uniform branches of the
argument on every open disc D C C*: remove from C a half-line not meeting
D, for example llL if D is contained in the half plane Re(z) > 0, a case to
which one can always reduce by a rotation around the origin. In this particular case, it is clear that, for zED, one can choose A(z) in the interval
] - 7r /2, 7r /2[. In the general case, one sees that if A(z) is a uniform branch
of the argument in D, then the set A(D) of values taken by A(z) for zED
is an interval of length < 7r.
(vii) Liftings of a path. By definition, a continuous path in C is a continuous map of a compact interval I = [a, b] of lR into C; this definition
extends trivially to any metric space, and in particular to the graph r of the
correspondence Arg introduced above. If t 1--+ 'Y(t) is a path in r, we have
(21.10)
'Y(t) = ("to(t),A(t»
where 'Yo(t) E C* and A(t) E lR are continuous functions of t; it is clear that
'Yo : I - + C* is a continuous path in C* and that, for all tEl, one must
have A(t) E Arg['Yo(t)], the set of possible arguments of 'Yo(t).
If, conversely, one has a continuous path ' Y in C* and chooses a number
A(t) E Arg!"f(t)] for all tEl so that the function A(t) is continuous, i.e. is a
uniform branch of the argument along'Y (similar definition to the case of .cog),
then t 1--+ ("t(t), A(t» is a continuous path in r which projects horizontally