Complex Analysis

Math 312  MWF 10:30-11:25am
Spring 1996  201 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1-3pm & Thu 1-2pm

Worksheet �5 (Monday 02/05/96)

SUMMARY square roots of complex numbers; algebraic/transcendental numbers

The primitive m-th root of unity, ω_{m} is de�ned as

ω_{m} = exp (\frac{2πi}{m}) =

The principal root of a complex number is If you look at the complete set of roots of unity you can see that they can be written as

1, ω_{m}, ω_{m}^{2}, ω^{3}, …, ω_{m}^{m−1}

We will show this for the four 4-th root of 1, 1^{1/4} below

c_{0} =  1
c_{1} =  ω_{m} =
c_{2} =  ω_{m}^{2} =
c_{3} =  ω_{m}^{3} =

We can also take the m-th roots of other numbers besides 1, say z_{0}. In that case one needs to solve

z^{m} = z_{0}

Write down the formula for {d_{k}}_{k=0}^{m−1}, the m m-th roots of z_{0} d_{k} =

Use your formula to �nd the cube roots of −8i.

Here’s another more challenging example: Find \sqrt{5 − 12i}.