24 1. The Complex Numbers

it

−it

−it

it

Figure 1.4.4. The Cut Line Discontinuities of

√

z2 + 1.

Note that for any function defined in terms of a branch log z of the log func-

tion, we can expect trouble along the cut line for log. Since, log has a jump or

discontinuity as we cross the cut line, we can expect the same for functions defined

in terms of it.

Also, we emphasize that functions defined this way depend on the choice of a

branch of the log function. If one wants such a function to agree with the standard

one on the positive real axis, then one must choose a branch of the log function

which is the ordinary natural logarithm on the positive real numbers. By Theorem

1.4.8, Part(e), this happens if and only if the interval I, in which arg(z) is required

to lie, contains 0. In particular, the principal branch has this property.

Exercise Set 1.4

1. Put each of the complex numbers −1, i, −i, 1+

√

3 i, and 5 − 5i in polar form.

2. Put each of the complex numbers e4πi, 3 e2πi/3, 5 e5πi/2, and 2 e−3πi/4 in stan-

dard form.

3. Find all powers of eπi/8. How many distinct powers of this number are there?

4. Show that (1 − i)7 = 8(1 + i) by converting to polar form, taking the seventh

power, and then converting back to standard form.

5. Using a calculator, calculate (1.2 e.5 i)n for n = 1, · · · , 6 and graph the resulting

points. Do the same for (.8 e.5 i)n.

6. Prove that if z is a number on the unit circle, then z has finitely many distinct

powers zn if and only if the argument of z is a rational multiple of 2π.

7. What are the 4th roots of unity?

8. What are the cube roots of −9?

9. Show that, given one nth root of z, the others are obtained by multiplying it

by the nth roots of unity.

10. Find argI z if (a) z = −i and I = (−π, π], (b) z = −i and I = [0, 2π), (c) z = 1

and I = [3π/2, 7π/2).