1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

164 CHAPTER 5 • ELEMENTARY FUNCTIONS

The domain for the function Log is the set of all nonzero complex numbers
in the z plane, and its range is the horizontal strip { w : -7f < Im ( w) ~ 7f} in the
w plane. We stress again that Log is a single-valued function and corresponds
to setting n = 0 in Equation (5-12). As we demonstrated in Chapter 2, the
function Arg is discontinuous at each point along the negative x-axis; hence so is
the function Log. In fact, because any branch of the multivalued function a.rg is
discontinuous along some ray, a corresponding branch of the logarithm will have
a discontinuity along that same ray.

• EXAMPLE 5.3 Find the values of log (1 + i) and Jog (i).

Solution By standard computations, we have

log(l + i) ={!nil + i i +i(Arg(l +i) + 2n7r): n is an integer}

= {In J2 + i (~ + 2n7r) : n is an integer} and

log(i) = {lnli l +i(Arg(i) + 2n7r): n is an integer}

= {i(i+2n7r) :nisaninteger}.

The principal values are

. rr. .7r ln2 .7r
Log(l+i) = lnv2+i4=--z+t4

L og ( i ") = i. 7i

2

.

and

We now investigate some of the properties of log and Log. From Equations

(5-10), (5-12), and (5-14), it follows that

exp (Log z ) = z for all z of; 0 and

Log(expz) = z, provided - 7f < Im(z) ~ ir,

(5-15)

(5-16)

and that the mapping w = Log(z) is one-to-one from domain D = (z: lzl > O}

in the z plane onto the horizontal strip {w: -ir <Im (w) ~ 7r} in thew plane.

The following example illustrates that, even though Log is not continuous

along the negative real axis, it is still defined there.

• EXAMPLE 5.4 Identity (5-14) reveals that

Log {- e) = In I-el+ iArg (- e) = 1 + i7r and

Log (-1) = In l- 11 + iArg (-1) = i7r.