Three representations of a cosine with amplitude a > 0, frequency w >= 0, and phase angle q:  
Real Trigonometric:  a cos(wt + q) 
Complex Exponential:  ½ a e^{jq}e^{jwt} + ½ a e^{jq}e^{jwt } 
Real Part of a Phasor:  Re {ae^{j(wt + q) }} 
A phasor can be viewed as a rotating vector in the complex plane that has length a and, at any time t, angle wt + q. The real part of a phasor is the projection onto the horizontal (real) axis. However for graphical demonstration it is more convenient to project onto the vertical axis. Therefore we rotate all phasors 90° in in the movie presentations and project on the vertical. In mathematical terms, we are using the relationship:

Three representations of a Fourier Series with fundamental frequency w:  
Real Trigonometric Sum:  Sum_{k=0,...,oo} a_{k} cos(kwt + q_{k}) 
Complex Exponential Sum:  Sum_{k=oo,...,oo} ½ a_{k} e^{jqk} e^{jkwt} 
Real Part of a Phasor Sum  Re { Sum_{k=0,...,oo} a_{k} e^{j(kwt + qk)}} 
Because projections on the vertical axis are easier to view than projections on the horizontal (real) axis, all phasors are rotated 90° in the movie presentations. 
Time Shift  
Time shift of a periodic signal corresponds to a phase shift of each phasor by an amount proportional to the phasor's frequency.  
Fourier Series Approximation of a Signal 
The Nharmonic Fourier series approximation x_{N }(t) of a signal x(t) is, in complex form: 
x_{N}(t) = Sum_{k=N,...+N} ½ a_{k} e^{jkwo t} 
where the coefficients a_{k} are given by: 
a_{k} = ( ^{1}/_{To}) _{To /}^{/} x(t) e^{jkwo t} dt

Convergence of the Fourier Series 
The coefficients of the Nharmonic Fourier series approximation minimize the integral squarederror over one period of the signal: 
E_{N} = _{To /}^{/}  x(t)  x_{N }(t)  ² dt 
Indeed, as the number of harmonics N is increased, the value of the integral squarederror converges to 0: 
lim_{N>oo}E_{N} = 0.

Windowing Techniques  
Windowing techniques provide attenuation factors for the k^{th} harmonic in an Nharmonic Fourier series approximation.  
Féjer Window:  (Nk)/N 
Hamming Window:  [ 0.54 + 0.46cos (pi × k/N) ] 
Note that for the k = N harmonic, the Féjer window yields a zeromultiplier while the Hamming window uses a 0.08 multiplier. For the DC term, both windows use a unity multiplier. 