ANSWERS 589- Make use of standard techniques. For example, to show that f + g is con-
tinuous, use T heorem 2.2 applied to the sum of two functions.
Section 2.4. Branches of Functions: page 84
la. The sector p > 0, ~ < </> < ~.le. The sector p > 0, 4" < </> < ~·
- Since f2,, (z) = r4 cos~ + ir4 sin~' where 21r < (} s; 41r (Explain!), we see
that the point (r4 cos~' r4 sin~) will lie in the lower half-plane (again,
explain). Thus, the range of fz,, (z) is {z: Im (z) s; 0, z 'f O}.
1 ·~. 3 ( 1 ·A<&(•) ) 3 ·A<•«>
Sa. Ji (z) = lzl^3 e' • , so U1 (z)) = lz l^3 e'-r- = lz l e' 3 _= z. This
shows that f 1 is indeed a branch of the cube root function.
7. The function f t (z) = r!ei~, where 0 'f z = rei^8 , and t < 0 s;^9 ,;' does
the job. Explain why, and find the range of this function, or of a different
function that you concoct.
·Ara:h~+2nk- For k = 0, 1 , 2. we have fk (z) = e' as the three b ranches of the
cube root with domains Dk = {z: z 'f O}. As in the text, slit each domain
along the negative real axis, and stack Do, D1, and D2 directly above each
other. Join the edge of Do in the upper half-plane to the edge of DJ in the
lower half-plane. Join the edge of D 1 in the upper half-plane to the edge
of D 2 in the lower half-plane. Finally, join the edge of D 2 in the upper
half-plane to the edge of Do in the lower half-plane. To really impress your
teacher, make a sketch or real 3D model of this!
Section 2.5. The Reciprocal Transformation: page 90- The circle Ci (-~i) = {w: lw+ ~ii=~}.
- The circle c~ (-!) = {w: lw +! I = n.
5. The circle C,/2(1-i) = {w: lw- 1 +ii= J2}.
7. The circle c~ (~) = {w: lw-~I= t}.
- Let e > 0 be given. Choose R =~·Suppose lz l > R. Then f.r < A= e, so
If {z) - 01 = l~I < e, i.e., f (z) E D,, (0).
11. The exterior of the disk D 1 {-~) = { (u, v): u^2 + (v + !)
2
> 1}.- The disk D,/2(1 - i) = { (u, v): (u - 1)^2 +(v+1)^2 < 2}.