1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5 .2 • THE COMPLEX LOGARITHM 163

(e) Use part (d} to show that p" (y) + p (y) = 0 and q" (y) + q (y) = O.

(f) Identify the general solutions to part (e). Then, given the initial condi-

tions f (0) = f (0 + Oi) = u (0, 0) +iv (0, 0) = 1 + Oi, find the particular

solutions and conclude that Identity (5--1) follows.

5.2 The Complex Logarithm

In Section 5.1, we showed that if w is a nonzero complex number, then the

equation w = exp z has infinitely many solutions. Because the function exp (z)

is a many-to-one function, its inverse (the logarithm) is necessarily multivalued.

I Definition 5.2: Mult ivalued logarithm

For z =/= 0, we define the multivalued function log as the inverse of the exponential

function; that is,

log(z) = w iff z = exp(w). (5-10) I

If we go through the same steps as we did in Equations (5-8) and (5-9), we

find that, for any complex number z =/= 0, the solutions w to Equation (5-10)

take the form

w = ln lzl + iB (z =/= 0), (5-11)

where BE arg (z) and ln lzl denotes the natural logarithm of the positive number

lzl. Because arg (z) is the set arg (z) = {Arg (z) + 2n?T : n is an integer}, we can

express the set of values comprising log ( z) as

log (z) = {ln lzl + i (Arg (z) + 2mr) : n is an integer}

= In lzl + iarg (z),

(5-12)

(5-13)

where it is understood that Identity (5-13) refers to the same set of numbers

given in Identity (5-12).

Recall that Arg is defined so that for z =/= 0, -?T < Arg (z) ::; ?T. We call any

one of the values given in Identities (5-12) or (5-13) a logarithm of z. Note that

the different values of log (z) all have the same real part and that their imaginary

parts differ by the amount 2n?T, where n is an integer. When n = 0, we have a

special situation.

I Definition 5.3: Principal value of the logarithm

For z =/= 0, we define Log, the principal value of the logarithm, by

Log(z) = ln lzl + iArg(z). (5-14)