*Michael Fowler*

We’ve seen that any complex number can be written in the
form _{}, where *r* is the
distance from the origin, and _{}* * is
the angle between a line from the origin to *z*
and the *x*-axis. This means that if we
have a set of numbers all with the same *r*,
but different _{}* *’s, such as _{}, etc., these are just different points on the circle with
radius *r* centered at the origin in
the complex plane.

Now think about a complex number that moves as time goes on:
_{}

At time *t*, *z*(*t*)
is at a point on the circle of radius *A*
at angle _{} to the *x*-axis.
That is, *z*(*t*) is going around the circle at a steady
angular velocity _{}. We can also write
this:

_{}

and see that the point *z*
= *x* + *iy* is at coordinates _{}

The angular velocity is _{}, the actual velocity in the complex plane is *dz*(*t*)/*dt*.

Let’s differentiate with respect to time:

_{}

*Exercise*: what are the *x* and *y* components of
this velocity regarded as a vector? Show
that it is perpendicular to the position vector. Why is that?

This differential equation has real and imaginary parts on both sides, so the real part on one side must be equal to the real part on the other side, and the same for imaginary parts. That gives

_{}

so differentiating the exponential is consistent with the standard results for trig functions.

Differentiating one more time,

_{}

Again going to the picture of a complex numbers as a
two-dimensional vector, this is just the acceleration of an object going round
in a circle of radius *A* at angular velocity _{}, and is just _{} towards the center of
the circle, the familiar _{} Thinking physics
here, this is the motion of an object subject to a steady central force.

But what if we just equate the real parts of both sides? That must be a perfectly good equation: it is

_{}

This is just the *x*-component
of the circling motion, that is, *it is
the “shadow” of the circling point on the x-axis*:

A simple animation of this diagram can be found here.

Forgetting for the moment about the circling point, and
staring at just this *x*-axis equation,
we see it describes the motion of a point having acceleration towards the
origin (that is, the minus sign ensures the acceleration is in the *opposite* direction to that of the point
itself from the origin) and the magnitude of the acceleration is proportional
to the distance of the point from the origin.

In fact, motion of this kind is very common in nature! It is called *simple harmonic motion*.

A simple standard example is a mass hanging on a
spring. If it is initially at rest, and
the string has length *L* (stretched
from its natural length to balance *mg*)
then if it is displaced a distance *x*
from that equilibrium position, the spring will exert an extra force -*kx* and the equation of motion will be

_{}

This is *exactly* the
equation of motion satisfied by the “shadow” on the *x*-axis of a point circling at a steady rate.

The general solution is _{}, where a possible phase *d* is included so that the point can be anywhere
in its oscillation at *t* = 0.