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 + · · · + cos =
1
2
+
sin (n + 1/2)θ
2 sin θ/2
.
Hint: Take the real parts of both sides in the first identity.
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