1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

5.2 • THE COMPLEX LOGARITHM 167

y v

i(a+ 2ir)

w= log,jz)

z=ew ia

Figure 5.4 The branch w = log 0 ( z) of tbe logru:ithm.

of 8 E arg ( z) that lies in the range a < 8 S: a + 27r, then the function log.,
defined by

log 0 (z) = In r + i8, (5-20)

where z = rei^6 f 0, and a < () s; a + 2n, is a single-valued branch of the loga-
rithm function. The branch cut for log 0 (z) is the ray {rei": r ~ O}, and each

point along this ray is a point of discontinuity oflog 0 (z). Because exp (log" (z)] =

z, we conclude that the mapping w = log°' (z) is a one-to-one mapping of

the domain lzl > 0 onto the horizontal strip { w : a < Im ( w) ::;: a + 2n}. If

a < c < d < a+ 2n, then the function w = log 0 (z) maps the set D =

{r ei^9 : a< r < b, c < e < d} one-to-one and onto the rectangle R defined by

R = {u +iv: Ina< u < lnb, c < v < d}. Figure 5.4 shows the mapping w =
log°' (z), its branch cut {rei°': r > O}, the set D , and its image R.
We can easily compute the derivative of any branch of the multivalued loga-

ri t hm. For a particular branch w = log" (z) for z = rei^6 # 0, and a: < e < a:+ 2n

(note the strict inequality for 8), we start with z =exp (w) in Equations (5-10)
and differentiate both sides to get

d d

1 = dz z = dz exp (log 0 (z))

d

= exp (log" (z)) dz log" (z)

d

= z dz log°' (z).

Solving for £. log" (z) gives

d 1

• log (z)=-, forz=re;^9 fO,anda<e<a+2n.
  dz " z