Well it is known that

Now make what do you get?

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- Apr 11th 2011, 03:07 PM #1

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- Apr 11th 2011, 03:11 PM #2

- Apr 11th 2011, 03:12 PM #3

- Apr 12th 2011, 12:26 AM #4
Let's suppose that Leonhard Euler has been a poet and not a mathematician, so that we ignore his formula. In that case the answer to Your question may derive from the solution of the differential equation...

(1)

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 ?...

Kind regards

- Apr 12th 2011, 12:56 AM #5
In fact, any complex number can be written as

so it is quite easy to see that

.

Since

where

where .

So , and since we know that when

.

Therefore .

Now by letting we have

.

This is considered to be the most beautiful equation known, because it links the five fundamental constants of mathematics.

- Apr 12th 2011, 02:17 AM #6

- Apr 12th 2011, 02:55 AM #7

- Apr 12th 2011, 03:01 AM #8

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- Apr 12th 2011, 05:07 AM #9

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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