If you represent a sinusoidal waveform in the
complex plane (i.e. x axis is the real axis;
y-axis is the imaginary axis) you end up with a
vector of magnitude 'r' which continuously
sweeps out a circle of given radius (with r =
Exp[jwt], in this case the radius is 1 - the
coefficient before the exponential.
With increasing time (t), the real
value of the complex exponential will generate a
cosine wave while the imaginary values will trace
out a sine wave.
In general the imaginary portion of the complex
exponential is not a physical observable, that is
to say it is nothing more than a mathematical
simplification. For example in solving
differential equations with sinusoidal driving
functions it is much easier to find trial
solutions using an exponential as the derivative
of an exponent is just another exponent. So
usually the imaginary part only exists on paper
while solving a problem, and then is dropped, as
the real portion of the solution is the only
meaningful part. But, there are some cases
where the imaginary part is useful (and actually
represents a physical concept).
The example I can think of is in optical
properties of dielectric materials (examples
include glass, semiconductors - optical fibers are
a good example). In this case a traveling
electromagnetic wave (light!) is represented by a
sinusoidal function as it passes through a
material, and while it travels through the
material the wave is attenuated, that is to say
that some of the energy is absorbed by the
material. In order to represent this
mathematically one adds an imaginary component to
the dielectric constant of the material (if you do
not know this is a property of the material that
describes the interaction of the light with the
atoms in the material); this would be
Exp[-jwt]: the negative sign results in a
decreasing exponential function. To actually
find the power lost in the material one must
square the field of the electromagnetic wave (the
function that describes the sinusoidal light wave)
and with (Exp[-jwt])2 we get a real
decaying exponential that nicely represents the
power lost to the material.
Well, that was a long one. Sorry to put you
through that, but your question is really a
complex matter! (Pun intended). If you need more
examples I suggest looking in a good E&M book,
there should be lots of examples with Exp[-jwt].
So good luck to you.
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