1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 297

The value of log z corresponding to k ~ 0 is called the principal value,

or the principal branch of the logarithmic function. We write

Wo = Logz = lnz + i Argz

with -7r < Argz:::; 7r. Note that if z = x > 0, then Logz = lnx, i.e., the

principal branch is that branch which reduces to the natural real logarithm

when z is real and positive.

In general, we shall write Wk to indicate the branch of the function corre-

sponding to a given integer kin (5.23-2). As noted in (1.15), from (5.23-1)

and (5.23-3) we have the identity

elog z = z

which holds regardless of the chosen value of log z. We point out that the
function w = log z is not defined at oo, since the equation ew = oo has no
solution (finite or infinite)..
If a # O, oo is a complex constant, and the exponential to the base a
is defined by the equation

(5.23-4)

it results in a multiple-valued function. It can be made single-valued by

taking

which.is called the principal value of the exponential, denoted p.v. az. For

a= e the latter reduces to the ordinary exponential function. Note that if

we apply (5.23-4) to the case a = e, we get

(5.23-5)

which is multiple-valued. Hence the ordinary exponential function, as de-
fined in Section 5.18, is only the principal branch of the multiple-valued
function (5.23-5). The use of the same symbol to denote both functions
will not lead to ambiguity because in what follows we shall use the nota-
tion ez (or expz) only to denote p.v. ez. The properties oflogz, which are
somewhat similar to those of lnz, were stated in Theorem 1.12.
To describe geometrically the mapping defined by w = log z, it suf-
fices to reverse the roles of the z- and w-planes in Fig. 5.23, cutting the
new z-plane along the negative real axis from 0 to oo, both endpoints be-
ing excluded. The upper edge 0 A of the cut corresponds with the upper
boundary v = 7r of the fundamental strip So (Fig. 5.42). However, the
lower edge OB corresponds with the upper boundary v = -7r of the strip

$_ 1 = {w: -37r < Imw:::; -7r}. Then it is clear that in order to obtain

the Riemann surface upon which the function w = log z is single-valued,