22 1. The Complex Numbers
1 2 0
1 2 0
Figure 1.4.3. Principal Branch of the Log Function on an Annulus.
Definition 1.4.7. Given a half-open interval I of length 2π on the line R, let
argI z, for z = 0, be the value of arg z that lies in the interval I. Then the function
defined for z = 0 by
log z = log |z| + i argI z
will be called the branch of the log function defined by I. In the special case where
I = (−π, π], this function will be called the principal branch of the log function.
The following properties of the various branches of the log function follow easily
from the definition and the properties of the exponential function. The proofs are
left to the exercises.
Theorem 1.4.8. If log is the branch of the the log function determined by an
interval I, then
(a) if z = 0, then
(b) if z ∈ C, then log
= z + 2πki for some integer k;
(c) if z, w ∈ C, then log zw = log z + log w + 2πki for some integer k;
(d) log 1 = 2πki for some integer k;
(e) log agrees with the ordinary natural log function on the positive real numbers
if and only if the interval I contains 0.
Suppose the interval I defining a branch of the log function has endpoints a
and b with a b. Then, since b − a = 2π, the polar coordinate equations θ = a
and θ = b define the same ray. This ray is called the cut line for this branch of the
logarithm. Observe that if z and w are two complex numbers with |z| = |w| 0
which are close to each other, but on opposite sides of the cut line – say with argI z
near a and argI w near b – then log w − log z is nearly 2πi. In other words, as we
cross the cut line moving in the clockwise direction, the value of log jumps by 2πi.
Thus, log is not continuous at points on the cut line. Later we will show that it is
continuous everywhere else.