416
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
a uniform branch of .cog will similarly be a function L : G ----+ C satisfying
(21.4')
z = exp[L(z)]
for all z E G.
The existence of such branches is obvious if one does not imp'ose any supplementary condition on them: choose at random a point of the graph of
the given correspondence lying on the vertical through each z ("axiom of
choice"). But to apply this procedure to the correspondence Arg z amounts
to attributing an argument to each complex number z =f. 0 by drawing lots;
utility zero. In this very case, and in all similar cases - the theory of holomorphic functions provides them ad libitum -, one looks to construct uniform
branches which are at least continuous functions of the variable, and even analytic when that is possible. In what follows, we always assume, implicitly,
and often explicitly, that the uniform branches in question are continuous.
(iii) Uniqueness up to 2krr of uniform branches. Given an arbitrary subset
G of C*, do there exist on G continuous uniform branches z f--t A(z) of the
correspondence Arg or, what amounts to the same on putting
(21.5)
L(z) = log Izl + i.A(z),
of .cog? The reply depends on G and is by no means obvious, as we shall
see. Another problem, which we can resolve now: how does one pass from one
uniform branch to another?
The answer depends on the fact that if A' and A" are two continuous
uniform branches of the argument in G then the function f(z) = [A'(z) -
A"(z)l/2rr is continuous and has values in IE. It will not escape anybody that
this kind of object is rarely met outside the case of constant functions. This
would be obvious if G were an interval oflR, since the image f(G) would then
be an interval (intermediate value theorem) contained in IE, so reducing to a
single point.
In our present case, it is the same if G is connected. Suppose, for simplicity
that G is arcwise connected, the only useful case in practice; this means that,
for all u, v E G, there exists a continuous path in G joining 'U to v, i.e.
a continuous map 'Y : I = [a, b] --+ G of an interval of lR into G such
that 'Y(a) = u and 'Y(b) = v (for the case of an open subset of C, look
again at nO 20 of Chap. II, where piecewise linear paths suffice). Now let A'
and A" be two uniform branches of the argument in G, and let us suppose
them equal at a particular point u E G - if not, subtract a suitably chosen
multiple of 2rr from A"(z) - and let us show that A'(v) = A"(v) for all
v E G. To do this, let us choose as above a path in G joining u to v and
put f(t) = {A'b(t)] - A"['Y(t)]}/2rr. We obtain a continuous function f on
I with values in IE, so constant; since it vanishes at t = a, it does so also at
t = b, qed 36 . In conclusion, on a connected set G c CoO two uniform branches
36 A better proof would be to extend the intermediate value theorem as follows:
if f is a continuous real function defined on a connected subset G of C, then
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
a uniform branch of .cog will similarly be a function L : G ----+ C satisfying
(21.4')
z = exp[L(z)]
for all z E G.
The existence of such branches is obvious if one does not imp'ose any supplementary condition on them: choose at random a point of the graph of
the given correspondence lying on the vertical through each z ("axiom of
choice"). But to apply this procedure to the correspondence Arg z amounts
to attributing an argument to each complex number z =f. 0 by drawing lots;
utility zero. In this very case, and in all similar cases - the theory of holomorphic functions provides them ad libitum -, one looks to construct uniform
branches which are at least continuous functions of the variable, and even analytic when that is possible. In what follows, we always assume, implicitly,
and often explicitly, that the uniform branches in question are continuous.
(iii) Uniqueness up to 2krr of uniform branches. Given an arbitrary subset
G of C*, do there exist on G continuous uniform branches z f--t A(z) of the
correspondence Arg or, what amounts to the same on putting
(21.5)
L(z) = log Izl + i.A(z),
of .cog? The reply depends on G and is by no means obvious, as we shall
see. Another problem, which we can resolve now: how does one pass from one
uniform branch to another?
The answer depends on the fact that if A' and A" are two continuous
uniform branches of the argument in G then the function f(z) = [A'(z) -
A"(z)l/2rr is continuous and has values in IE. It will not escape anybody that
this kind of object is rarely met outside the case of constant functions. This
would be obvious if G were an interval oflR, since the image f(G) would then
be an interval (intermediate value theorem) contained in IE, so reducing to a
single point.
In our present case, it is the same if G is connected. Suppose, for simplicity
that G is arcwise connected, the only useful case in practice; this means that,
for all u, v E G, there exists a continuous path in G joining 'U to v, i.e.
a continuous map 'Y : I = [a, b] --+ G of an interval of lR into G such
that 'Y(a) = u and 'Y(b) = v (for the case of an open subset of C, look
again at nO 20 of Chap. II, where piecewise linear paths suffice). Now let A'
and A" be two uniform branches of the argument in G, and let us suppose
them equal at a particular point u E G - if not, subtract a suitably chosen
multiple of 2rr from A"(z) - and let us show that A'(v) = A"(v) for all
v E G. To do this, let us choose as above a path in G joining u to v and
put f(t) = {A'b(t)] - A"['Y(t)]}/2rr. We obtain a continuous function f on
I with values in IE, so constant; since it vanishes at t = a, it does so also at
t = b, qed 36 . In conclusion, on a connected set G c CoO two uniform branches
36 A better proof would be to extend the intermediate value theorem as follows:
if f is a continuous real function defined on a connected subset G of C, then