420
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
choice is unique up to the addition of a constant multiple of 27r, but, once
again, the argument of -yet) thus chosen depends on t and not only on -yet).
(ix) Uniform bmnches and closed paths. Let G be an open connected
subset of C* and assume that there exists a uniform branch A(z) of the
argument on G. Choose an a E G, put A(a) = a and, for z E G given, join a
to z by a continuous path -yet) where t varies in a compact interval I. The
function t ...... Ab( t)] is clearly a uniform branch of the argument along -y, the
branch whose initial value is a and final value is A(z). In particular, the final
value does not depend on the path -y chosen, and since every other uniform
branch of the argument along -y differs from the preceding by a constant
multiple of 27r, the result persists in the following form: if one "follows by
continuity" the value of the argument along a path joining a to z in G, the
value obtained at z depends only on the value chosen at a; in other words,
the "variation of the argument" along a path in G depends only on its origin
and on its end point and, in particular, is zero along every closed path, Le.
one whose end points are equal.
If, conversely, this condition is satisfied, one can construct a uniform
branch A(z) of the argument in G directly. Choose one of the possible of
values A(a) at a point a E G arbitrarily, then determine A(z) by joining a
to z by a continuous path in G: there exists a uniform branch of the argument
along this path, one can assume that its initial value is A(a) and A(z) is then,
by definition, its final value. The result being by hypothesis independent of
the path chosen in G, this is a good definition. To show that the result is
a continuous function of z, one chooses (figure 3) an open disc D c G with
centre z and, on D, a uniform branch A' of the argument according to the
result obtained in (vi); one can clearly assume A'(z) = A(z). This said, let z'
be a point of D; to join a to z', one can join a to z by a path -y : [u, v] ---+ G
and follow this path by a path -y' : [v, w] ---+ D joining z to z'; one clearly
obtains a uniform (Le. continuous) branch of the argument along the path
joining a to z' by insisting that it be equal to Ab(t)] along -y and to A'[-y'(t)]
along -y'. The final value being A'(z'), one thus has A(z') = A'(z') on D,
whence the continuity of A on G.
In consequence, for there to exist a uniform bmnch of the argument in G
it is necessary and sufficient that, for every a, bEG and path -y joining a to b
in G, the value at b of the argument of z obtained with the help of a uniform
bmnch of the argument along -y depends only on its value at a, and not on
the path followed; or again: that the "variation of the argument along any
closed path" in G be zero.
It is "geometrically obvious" that, for this to be so, it is necessary and
sufficient that no closed path traced in G surrounds the point OJ no matter
what mechanic or physicist explains to you that if you "follow the argument
by continuity" of a point z whose trajectory "surrounds" (in one sense or the
other) the origin once or twice, at the end you obtain a value of the argu-
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
choice is unique up to the addition of a constant multiple of 27r, but, once
again, the argument of -yet) thus chosen depends on t and not only on -yet).
(ix) Uniform bmnches and closed paths. Let G be an open connected
subset of C* and assume that there exists a uniform branch A(z) of the
argument on G. Choose an a E G, put A(a) = a and, for z E G given, join a
to z by a continuous path -yet) where t varies in a compact interval I. The
function t ...... Ab( t)] is clearly a uniform branch of the argument along -y, the
branch whose initial value is a and final value is A(z). In particular, the final
value does not depend on the path -y chosen, and since every other uniform
branch of the argument along -y differs from the preceding by a constant
multiple of 27r, the result persists in the following form: if one "follows by
continuity" the value of the argument along a path joining a to z in G, the
value obtained at z depends only on the value chosen at a; in other words,
the "variation of the argument" along a path in G depends only on its origin
and on its end point and, in particular, is zero along every closed path, Le.
one whose end points are equal.
If, conversely, this condition is satisfied, one can construct a uniform
branch A(z) of the argument in G directly. Choose one of the possible of
values A(a) at a point a E G arbitrarily, then determine A(z) by joining a
to z by a continuous path in G: there exists a uniform branch of the argument
along this path, one can assume that its initial value is A(a) and A(z) is then,
by definition, its final value. The result being by hypothesis independent of
the path chosen in G, this is a good definition. To show that the result is
a continuous function of z, one chooses (figure 3) an open disc D c G with
centre z and, on D, a uniform branch A' of the argument according to the
result obtained in (vi); one can clearly assume A'(z) = A(z). This said, let z'
be a point of D; to join a to z', one can join a to z by a path -y : [u, v] ---+ G
and follow this path by a path -y' : [v, w] ---+ D joining z to z'; one clearly
obtains a uniform (Le. continuous) branch of the argument along the path
joining a to z' by insisting that it be equal to Ab(t)] along -y and to A'[-y'(t)]
along -y'. The final value being A'(z'), one thus has A(z') = A'(z') on D,
whence the continuity of A on G.
In consequence, for there to exist a uniform bmnch of the argument in G
it is necessary and sufficient that, for every a, bEG and path -y joining a to b
in G, the value at b of the argument of z obtained with the help of a uniform
bmnch of the argument along -y depends only on its value at a, and not on
the path followed; or again: that the "variation of the argument along any
closed path" in G be zero.
It is "geometrically obvious" that, for this to be so, it is necessary and
sufficient that no closed path traced in G surrounds the point OJ no matter
what mechanic or physicist explains to you that if you "follow the argument
by continuity" of a point z whose trajectory "surrounds" (in one sense or the
other) the origin once or twice, at the end you obtain a value of the argu-