This is a very terse review of their most important properties. From
the figure above, we can see that an arbitrary complex number
can
*always* be written as:

where , , and .

There are a variety of ways of deriving or justifying the exponential form. Let's examine just one. If we differentiate with respect to in the second form (66) above we get:

(68) |

This gives us a differential equation that is an *identity* of
complex numbers. If we multiply both sides by
and divide both
sizes by
and integrate, we get:

(69) |

If we use the inverse function of the natural log (exponentiation of both sides of the equation:

(70) |

where is basically a constant of integration that is set to be the

There are a number of really interesting properties that follow from the
exponential form. For example, consider multiplying two complex numbers
and
:

(71) | |||

(72) | |||

(73) |

and we see that multiplying two complex numbers multiplies their