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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
. All complex numbers can be written as a real amplitude
times a complex exponential form involving a phase angle. Again,
it is difficult to convey how incredibly useful this result is without
devoting an entire book to this alone but for the moment, at least, I
commend it to your attention.
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:
where
is basically a constant of integration that is set to be the
magnitude of the complex number (or its modulus) where the
complex exponential piece determines its complex phase.
There are a number of really interesting properties that follow from the
exponential form. For example, consider multiplying two complex numbers
and
:
and we see that multiplying two complex numbers multiplies their amplitudes and adds their phase angles. Complex multiplication
thus rotates and rescales numbers in the complex plane.
Next: Trigonometric and Exponential Relations
Up: Complex Numbers and Harmonic
Previous: Complex Numbers and Harmonic
Contents
Robert G. Brown
2011-04-19