Complex Analysis

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

Worksheet �3 (Friday 01/16/98)

SUMMARY Polar and Exponential forms of Complex Numbers

READING Churchill & Brown, pages 12-18

We have been considering the complex plane as an analogue to the 2-D cartesian x-y plane. You may recall that there are other coordinate systems that can be imposed on the 2-D plane. One of those coordinate systems is known as polar coordinates. \underline{Exercise} Write (1, \sqrt{3}) in polar coordinates.

What is the angle between a line drawn from (0, 0) to (1, \sqrt{3}) and the x-axis?

The complex number which is found at (1, \sqrt{3}) is . It can also be written as z = r(cos θ + i sin θ) = |z| cis θ This is known as the polar form of the complex number. This angle θ corresponding to the complex number z is called the principal argument of z, denoted by Arg z. By tradition, −π < Arg z ≤ π and Arg 0 is unde�ned.

The set of all values θ of such that tanθ = \frac{Im z}{Re z} is known as the argument of z, or arg z

arg z = Arg z + 2nπ  (n = 0, ±1, ±2, …)  (1)

\underline{Exercise} Therefore, arg (1 + \sqrt{3}i) is .

Write z_{1} = −1 − i and z_{2} = 1 − i in polar form and sketch them on the graph below.

Do you see any interesting geometrical relationship between z_{1} and z_{2}?

Do you notice any interesting algebraic relationship between z_{1} and z_{2}?

Write down a general relationship between arg(z_{1}z_{2}) and arg(z_{1}) and arg(z_{2})

Exponential Forms

Euler’s Formula You may have noticed that the expression cos θ + i sin θ is so common that the shorthand cis θ has been developed. Another, even more compact symbol can be derived using Euler’s formula:

eiθ = cos θ + i sin θ

Note θ = arg z and is a real number in these formulas. So, now we can write complex numbers in exponential form

z = |z|eiθ = reiθ

\underline{Exercise}

1. Compute the value of e^{πi}

2. Write 1 + i in exponential form

3. Write (1 − i)^{5} in rectangular form (i.e. the usual x + iy)

De Moivre’s Formula

(cos θ + i sin θ)^{n} = cos nθ + i sin nθ

We will prove this is true by �rst writing down z^{n} in polar form and exponential form and equating them.