Well it is known that
Now make what do you get?
The (1) is the trajectory of a point the velocity of which is its postion rotated by so that we have an 'uniform circular move'. The solution of (1) is so that the question is: where will be the point at the time ?...
so it is quite easy to see that
So , and since we know that when
Now by letting we have
This is considered to be the most beautiful equation known, because it links the five fundamental constants of mathematics.
there are many ways of "deriving" this truth. some require you to have some knowledge of complex numbers, or calculus, upon which a sort of proof is based.
i, however, will not do this. instead, i will try to give you a sense of what e^z MEANS, when z is a complex number.
one can write a complex number as a two-part sum, one component in the "x" or "real" direction, and another in the "y" or "imaginary" direction. so we can write z as follows:
z = x + iy. now if we assume that exponents obey some of the rules we have come to expect in the real case, we ought to have:
e^z = e^(x+iy) = (e^x)(e^iy).
and so far, so good, and the e^x part behaves as expected, as x gets bigger, it gets bigger. as x becomes very negative, it tends toward 0.
but the overall effect is not to transform the plane into some inverted parabola-like shape, as you might expect. and this is because of the e^(iy) part.
if you hold x constant, and travel along the y-direction, e^(iy) traces out a circle in the complex plane. up-and-down, rather than being strictly increasing,
e^(iy) is periodic, like a trig function. so while on the left-hand side of the y-axis, e^z is rather flat, on the right-hand side it increases rapidly to the left in amplitude,
and undulates up and down as you travel either up, or down, in a delicate balance of interacting sines and cosines. along the vertical line x = 0,
e^(iy) is bending the y-axis into a circle of radius 1, starting anew every 2 (which is the period).
and so it happens, that at y = , we are half-way 'round a circle-cycle, and the value of the combined coordinates is -1 + 0i