To introduce the use of the complex representation as a powerful
method for the analysis of problems with cyclic behavior, we shall
solve a problem first using the traditional approach and then using
complex analysis. For the problem, let us ask for the amplitude A
and initial phase 
of a simple harmonic oscillator with
initial conditions 
and 
. Note that the traditional
analysis here will get somewhat involved. What we wish to underscore
here is that nothing mysterious underlies the complex approach but
that it does make the analysis significantly simpler once you become
used to it.