O que eu adoro esta demonstração

Here is the well known identity that e^(ix) = cos(x) + i.sin(x)

Then let x = pi.  cos(pi) = -1,  sin(pi) = 0
so     e^(i.pi) = cos(pi) + i.sin(pi)
              = -1 + 0  
and so e^(i.pi) = -1

From this you can also write the FAMOUS FIVE equation connecting the
five most important numbers in mathematics, 0, 1, e, pi, i

         e^(i.pi) + 1 = 0

To show the truth of the identity quoted above we let
z = cos(x) + i.sin(x)
Then dz/dx = -sin(x)+i.cos(x)
         = i{cos(x)+i.sin(x)}
         = i.z

So  dz/z = i.dx   Now integrate
 ln(z) = i.x + const.    When x=0, z=1 so const=0
 ln(z) = i.x

    z = e^(i.x)

So cos(x) + i.sin(x) = e^(i.x)

-Doctor Anthony,  The Math Forum.