1C (Vectors and Matrices)

z^{i}

has the form

exp (i (lo g ∣ z ∣ + i (ar g z + 2 πk))) .

For this to equal one, its exponent must be

2 πin

for some

n \in Z

. Hence a logarithm of

z

must equal

2 πn

, so

z

is positive real and

z = e^{2 πn}, n \in Z .

With the principal branch this is the same list; in the multivalued convention these are precisely the

z

for which one value of

z^{i}

equals one.

b

The circle is

∣ z - 5 i ∣ = 5 .

The direction from its center

c = 5 i

z_{0} = 15/4

z_{0} - c = \frac{15}{4} - 5 i, ∣ z_{0} - c ∣ = \frac{25}{4},

so the corresponding unit direction is

3/5 - 4 i /5

. Moving a distance five from the center in the two directions gives

z = 5 i \pm 5 (\frac{3}{5} - \frac{4}{5} i) .

Thus the intersections are

z = 3 + i and z = - 3 + 9 i .

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