1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.4 • BRANC HES OF FUNCTIONS 81

so f 1 and h can be thought of as "plus" and "minus" square root functions.
T he negative real a.xis is called a branch cut for the functions f 1 and h Each
point on the branch cut is a point of discontinuity for both functions j 1 and h

• EXAMPLE 2.21 Show that the function Ji is discontinuous along the neg-
  ative real axis.

Solution Let z 0 = r 0 ei" denote a negative real number. We compute the limit
as z approaches zo through the upper half-plane {z : Im (z) > O} and the limit as
z approaches zo through the lower half-plane {z: Irn(z) < O}. In polar coordi-
nates these limits are given by

I1m. f 1 ( re i9) = lim r• l ( cos^8 - +istn.. -8) =ir. 0 ~ ,

(r,9)-(ro,w) (r,9)-(ro,,,.) 2 2

and

I1m. f 1 ( re •9) = 1· rm r • l ( cos-f).. f)).!

2

+ism-

2

=-ir 0.

(r,9)-(ro.-w) (r,9) -(ro,-11')

The two limits are distinct, so the function Ji is d iscontinuous at zo.

R e mark 2.4 Likewise, h is discontinuous at zo. The mappings w = !1 (z),
w = h (z), and the branch cut are illustrated in Fig ure 2.18. •

We can construct other branches of the square root function by specifying

that an argument of z given by e = arg z is to lie in the interval a < e ~ a + 27!'.

y v

w=f.(z)

u

• i

y

w=fl(z)

x ---- u

-i

Figure 2.18 The branches / 1 and f2 off (z) = z!.