1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

10.3 • MAPPINGS INVOLVING ELEMENTARY F UNCTIONS 4 1 5

where r1 = lz - l j and 81 = Arg (z - 1), and

l ·~
(z + 1 )2 = .ji2e' • ,

where r2 = lz + l j and 82 = Arg(z + 1).

The discontinuities of Arg (z - 1) and Arg (z + 1) are points on the real axis
such that x::; 1 and x s -1, respectively. We now show that ft (z) is continuous

on the ray x < -1, y = 0.

We let z 0 = x 0 + iy 0 denote a point on the ray x < -1, y = 0, and then

obtain the following limits as z approaches zo from the upper and lower half-
planes, respectively:

lim fi(z) = lim Jrlei!.f lim .ji2ei!t
z-zo, lmz>O r1-lxo-ll. 81 - n r r2-lxo+ l l, 02~1f

and

= Jlxo - l l (i) J lxo + ll (i)

= -Jlxfi- 11

= Jlxo - l l (-i) J lxo + ll (-i)

= -Jlxfi-11.

Both limits agree with the value of f 1 (z 0 ) , so it follows that / 1 (z) is continuous

along the ray x < -1, y = 0.

We can easily find the inverse mapping and express it similarly:

1 1 l

z = gi( w) = ( w^2 + 1) • = ( w + i)2 ( w - i)^2 ,

where the branches of the square root function are given by

1 ·ll

(w + i)^2 =,,/Pie'> ,

where p 1 = lw +ii, t/> 1 = arg=r (w + i), and - 2 " < arg-T (w + i) <^3 ;, and

(w - i)t = JP2ei~,

where f>2 = lw - ii, t/12 = arg-; (w -i), and - 2 " < arg=r (w - i) <^3 ;.

A similar argument shows that g 1 (w) is continuous for all w except those

points that lie on the segment u = 0, - 1 S v S 1. Verification that

91 (!1 (z)) = z and! 1 (9 1 (w)) = w

hold for z in D 1 and w in H 1 , respectively, is straightforward. Therefore, we

conclude that w =Ji (z) is a one-to-one mapping from D 1 onto H 1. Verifying

that f 1 (z) is a lso analytic on the ray x < - 1, y = 0, is tedious. We leave it as a

challenging exercise.