1.4. Polar Form for Complex Numbers 25

11. For the principal branch of the log function, find log(1 − i).

12. Find log(1 − i) for the branch of the log function determined by the interval

[0, 2π).

13. Prove (a) and (b) of Theorem 1.4.8.

14. Prove (c), (d), and (e) of Theorem 1.4.8.

15. Analyze the function zi defined by (1.4.7) using the principal branch of the log

function. What kind of a jump, if any, does it have as z crosses the negative

real axis?

16. Analyze the function

√

1 − z2, where the square root function is defined by

the principal branch of the log function. Where does it have discontinuities

(jumps)?

17. Let the square root function be defined by the principal branch of the log

function. Compare the functions

√

z2 − 1 and

√

z + 1

√

z − 1. Where are the

discontinuities of each function?

18. The identity 1 + z +

z2

+ · · · +

zn

=

1 −

zn+1

1 − z

was derived in Example 1.2.7.

Use this to derive Lagrange’s trigonometric identity:

1 + cos θ + cos 2θ + · · · + cos nθ =

1

2

+

sin (n + 1/2)θ

2 sin θ/2

.

Hint: Take the real parts of both sides in the first identity.