1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.2 • THE COMPLEX LOGARITHM 165

When z = x + iO, where x is a positive rea.1 number, the principal value of
the complex logarithm of z is

Log (x + iO) = lnx + iArg (x) = lnx + iO = lnx,

where x > 0. Hence Log is an extension of the real function In to the complex
case. Are there other similarities? Let's use complex function theory to find the

derivative of Log. When we use polar coordinates for z = rei^8 =/= 0, Equation

(5-14) becomes

Log (z) = In r + iArg (z)

= In r + i8, for r > 0 and - 'Tf < 8 <.!, 7r

= U (r,8) + iV (r,B),

where U (r, 8) = In r and V (r, 8) = 8. Because Arg (z) is discontinuous only at

points in its domain that lie on the negative real axis, U and V have continuous

partials for any point (r, 8) in their domain, provided rei^9 is not on the negative

real axis, that is, provided -7r < 8 < 7r. (Note t he strict inequality for 8 here.) In

addition, the polar form of the Cauchy- Riemann equations holds in this region

(see Equation (3-22) of Section 3.2), since

1 1 - 1

U, (r, 8) = -Ve (r, 8) = - and Vr (r, 8) = - Ue (r, 8) = 0.

r r r

Using Theorem 3.5 of Section 3.2, we see that

-Log(zd ) = e- •.^9 (U, +iV,) = e- •.^9 (1 - + Oi ) 1 1 = -. = -

dz r re•e z'

provided r > 0 and -7f < B < 7r. Thus, the principal branch of the complex

logarithm has the derivative we would expect. Other properties of the logarithm

carry over, but only in specified regions of the complex plane.

• EXAMPLE 5.5 Show t hat the identity Log (z 1 z 2 ) = Log (z 1 ) + Log (z 2 ) is
  not always valid.

Solution Let z 1 = -./3 + i and z2 = - 1 + i./3. Then

Log (z1 z2) = Log ( - 4i)

= ln4+i (-i), but

.51f. 27r

Log(z 1 ) + Log(Z2) = ln2 + i

6

+ ln2 + i

3

1

.31f

= n4+ i

2

.