Complex numbers in Polar Form

Euler's Equation

We now will introduce a remarkable relationship that has fascinated mathematicians ever since its discovery in the 18th century.

It is known as Euler's equation.

The relationship says that

There are many ways to prove this. One method involves series and is perhaps the most convincing, but we will take a different

approach. In order to show that this is true, we will consider a differential equation.

Consider the following derivatives:

Now consider the following differential equation:

where derivatives are taken with respect to

. What function satisfies this statement?

Based on our above derivatives, we can see that the only candidates are

and and and

We now invoke a theorem from the study of differential equations that says that if more than two functions are all solutions of a

second-order equation, then any one of them must be a linear combination of the other two, so that

Dividing all by we get

where

and are constants.

To find these constants, we first plug a number into this statement. The number

we first take the derivative (with respect to )

and plug in

again:

.

We're done:

You could show similarly that .

This theorem allows us to write any complex number in a simple way using only its modulus and argument.

For a complex number

we get

This is referred to as the polar form or polar representation of the number

.

Purple Question: Prove that

algebraically, then draw a diagram showing why this is true geometrically using similar

triangles.

Violet Question: Draw a complex number on the plane whose modulus is larger than 1. Now show geometrically where the

number

is located.

Skill Set

Complex Numbers in Polar Form

No. 1

Blue Question: (P. Zeitz) Deduce the following result using only geometry and without doing any calculation:

STEP 2009, Math III, #6: Show that

for . Hence show that

where .

Interpret this result as a theorem about cyclic quadrilaterals. [Note, you may find it helpful to use a few trig identities.]

As a side note for purists: Note that I haven't really discussed what on earth we could possibly mean by a real number raised to an

imaginary power. This is perhaps stretching the concept of an exponent too far. In fact, for a really satisfying discussion of what

this might mean, we will need to refer to series. You may refer on your own to Visual Complex Analysis by Needham.

Complex Numbers as Rotations

Consider a complex number . Notice what happens when we multiply this number by another complex number ,

with argument .

and

Green Question: (Needham VCA) Just by considering the product

, show that

and , for .

Blue Question: (P. Zeitz) Derive the tangent double-angle identity

, , and is equilateral if and only if

STEP 1998, Math I, #5: (i) In the Argand diagram the points Q and A represent the complex numbers

, , and be real numbers. Find a condition of the form , where , , and are integers, which

ensures that

, , and represent the vertices of an equilateral triangle

(in clockwise or anti-clockwise order) if and only if

Show that the roots of the equation

, where and are given complex numbers with , the equation (*) becomes

, and respectively, where

, and are positive real numbers. Sketch the triangle ABC.

Three equilateral triangles ABL, BCM and CAN (each lettered clockwise) are erected on sides AB, BC and CA respectively. Show that

the complex number representing N is

and find similar expressions for the complex numbers representing L and M.

Show that lines LC, MA and NB all meet at the origin, and that these three line segments have the common length

.

STEP 2001, Math II, #7: In the Argand diagram, O is the origin and P is the point

. The points Q, R, and S are such that the

lengths OP, PQ, QR, and RS are all equal, and the angles OPQ, PQR, and QRS are all equal to

, so that the points O, P, Q, R, and

S are five vertices of a regular 12-sided polygon lying in the upper half of the Argand diagram. Show that Q is the point

and find S.

The point C is the centre of the circle that passes through the points O, P, and Q. Show that, if the polygon is rotated anticlockwise

about O until C first lies on the real axis, the new position of S is

.

STEP 2007, Math III, #6: The distinct points P, Q, R and S in the Argand diagram lie on a circle of radius

centred at the origin and

are represented by the complex numbers

, , , and respectively. Show that

Deduce that, if the chords PQ and RS are perpendicular, then .

The distinct points where lie on a circle. The points lie on the same circle and are chosen so that the

greater than 4.

STEP 2005, Math III, #8: In this question,

and , respectively. The point P has coordinates . The perpendicular

, , and representing A, B, and C satisfy

, , and is the point represented by

, , , and have unit modulus, and arguments , , and ,

respectively, with

.

Show that

where

or 1. Deduce that

[You may find some of the identities on this page useful.]

Complex functions as mappings

Complex numbers require two dimensions to be visualized fully. So a function that takes complex numbers as inputs and gives

complex numbers as outputs would require four dimensions to be graphed onto a single set of axes! This is clearly impossible,

so we need other methods to visualize complex functions. One of the most common methods is to use side-by-side grids to show

what a function does to individual small regions of the plane.

The input region is called the locus of the inputs values and the output region is called the locus of the function.

For a simple example, consider the mapping or complex function

but where the locus of

is the modulus of and is the argument of .

If then . Furthermore, if then and if then . Therefore input values lying in the first

quadrant inside the circle of radius 1 will get mapped onto output values lying in the upper half-plane within the circle of radius 1.

Input

maps the entire complex plane onto itself twice. One way to visualize this is that the complex plane gets cut at the positive real axis

then stretched around

to form a sort of two-level parking garage.

STEP 2011, Math III, #8: The complex numbers

and are related by

, , and are real. Express and in terms of and .

describe the same locus in the complex z-plane. Sketch this locus.

(ii) Prove that the equation

(**)

describes part of this same locus, and show on your sketch which part.

(iii) The complex number

is related to by

.

Determine the locus produced in the complex

-plane if satisfies (*). Sketch this locus and indicate the part of this locus that

corresponds to (**).

Sine and Cosine as Complex functions

Euler's equation

has already been used to obtain an elegant representation for any complex number. However, we can also use it to look at the trig

functions in a different way. First we observe that, since sine is an odd function and cosine is an even function,

Combine these two equations to get

and

Pause for a second to consider what this says. Those trigonometric functions that you've studied since Algebra II days based on

the unit circle and originating in similar right triangles...are really exponential functions when viewed through the lens of complex

numbers!!!

Consider the following simple algebraic proof that uses complex numbers but which says something about the real trig functions.

Skill Set

Complex Numbers in Polar Form

No. 2