1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 25

numbers other than -1 can be put into one-to-one correspondence with

the real numbers.

8. If u and v are unimodular, show that (u+v)/(l+uv) (uv i--1) is real.

9. If lzkl < 1, rk 2'.: O, and E~=l rk = 1, prove that

10. If IPI < 1 and lql < 1, prove that:

(a) I p-q I< 1 (b) IPI -lql

1 -pq 1 - lpllql I

p-q I

~ 1-pq

11. Show that

-- =1

I

p-qi

1-pq

if either IPI = 1 or lql = 1, or if IPI = lql = 1 provided that p i-q.

12. Prove the identity

IP+ ql^2 + IP -ql^2 = 2[IPl^2 + lql^2 l

In particular, if IPI = lql, then

IP+ ql^2 + IP -ql^2 = 4IPl^2

13. Prove the identity

lz11^2 + lz2 l^2 + l;a 12 + lz1 + z2 + za 1

2

= lz1 + z2 l

2

• h + za 1

2

+ lza + Z11

2

14. Prove the identity
    11 - ab12 -la -bl2 = (1 + labl)2 -(lal + lbi)^2

15. If Re a > 0 and Re b > O, show that

la -bl < la+ bl

16. Prove the identity
    .

(a - b)(l +cc)= (a - c)(l +be)+ (c - b)(l + ac)

*17. Show that

(1 + zw)(l + zw) ~ (1 + lzl^2 )(1 + lwl^2 )

and use it together with the identity in Problem 16 to prove that

la -bl(l + lcl2)1/2 ~ la -cl(l + lbl2)1/2 + le - bl(l + lal2)1/2

18. Show that