Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Class �4 (Wednesday 01/21/98)

SUMMARY Polynomial Equations of a Complex Variable and Roots of Complex Numbers CURRENT READING Churchill & Brown, pages 19-23 NEXT READING Churchill & Brown, pages 23-32 \underline{Update} The answers to the exercises from the class �3 are: 1: eiπ = cos π + i sin π = −1 2: 1 + i = |1 + i|ei arg (1 + i) = \sqrt{2}e^{\frac{iπ}{4}+2niπ}

3: (1−i)^{5} = (|1 − i|ei Arg (1 − i))^{5} = (\sqrt{2}e^{i \frac{−π}{4}})^{5} 2^{5/2}e^{\frac{−i5π}{4}} = \sqrt{32}e^{\frac{3πi}{4}} = \sqrt{32}\frac{(i−1)}{\sqrt{2}} = −4+4i

We now know how to deal with real integer powers of complex numbers in a nice way (by using DeMoivre’s Formula and exponential form).

Fractional Exponents

What about real fractional powers of complex numbers, i.e. roots? That is, we want to solve an equation like

z^{n} − z_{0} = 0  (1)

where z_{0} is a known complex number, and we are trying to �nd the corresponding value(s) of z = z_{0}^{1/n} which solve this equation. Suppose we write z_{0}, z and z^{n} in polar form:

z_{0} =  (2)
z =  (3)
z^{n} =  (4)

where |z_{0}| = r_{0}, |z| = r, Arg z_{0} = θ_{0} and Arg z = θ We can rewrite (1) as z^{n} = z_{0} and obtain two real equations.

\underline{Exercise} Obtain expressions for r and θ in terms of r_{0} and θ_{0}

Roots of Unity

We are interested in �nding the n^{th} roots of unty, i.e. z such that

z^{n} = 1, n = 1, 2, 3 …

On the following axes, draw vector representations of the n^{th} roots of unity when n = 2, n = 3 or n = 4. How many distinct solutions to z^{n} = 1 are there? In other words, we are trying to evaluate \sqrt{1}, ^{3} \sqrt{1}, ^{4} \sqrt{1}, …

What do you think the 5^{th} roots of unity will look like? \underline{Exercise} Compute the solutions to the equation z^{5} = i and write them in polar (and rectangular) form. Sketch these solutions on the grid on the right.

Using DeMoivre’s Formula and the result on n^{th} roots we can obtain a general formula for evaluating z^{m/n}

z^{m/n} = c_{k} = |z|^{m/n} exp (\frac{i(θ + 2πk)}{n}), k = 0, 1, …, n−1, where θ = Arg z

These n roots can be written as

c, cω_{n}, cω_{n}^{2}, cω_{n}^{3}, ⋯, cω_{n}^{n−1}

where c is any n^{th} root of a non-zero complex number, and ω_{n} = exp (\frac{2iπ}{n} )

\underline{Examples}

(1) Prove that sin 2θ = 2 sin θ cos θ and cos 2θ = cos^{2} θ − sin^{2} θ (use DeMoivre’s Formula)

(2) Evaluate \sqrt{5 − 12i}

(3) Solve w^{3} − i = −\sqrt{3}

(4) Solve w^{4/3} + 2i = 0