The napierian logarithmic base e is also a special constant. The function y=e^{x} is the natural
exponential function and has interesting properties. It is the *only function* that is it's own derivative
and it's own integral. IF: y=e^{x}, then: y'=e^{x}, y"=e^{x}, etc. If x=1, y=e and the
slope of y=e^{x} is also e and the rate of change of slope at x=1 is also e. Who made it work that way?
Also, the integral of e^{x}dx=e^{x}. Kudos for creation.

### 1) SUMmation: e=SUM[1/(k!)] for k=0 to infinity Created: 5-Jan-2011 2:00 PM

### 2) VALuation: e=(1+1/k)

^{k}as k approaches infinity Created: 5-Feb-2011### 3) Function: y=x

^{x}Created: 5-Feb-2011### 4) Function: e

^{(i*PI)}=-1 Created: 5-Feb-2011

The above summation very rapidly converges to the constant 'e' in 15 steps to 11 places, e=2.71828182846...

This function also converges to 'e' in one step as k approaches infinity. My laptop yields 2.7182818 for k=45E+6, math seems to run out of exponentiation accuracy for k.45E+6.

This function has a value approaching y=1 at x=0 and y=1 at x=1, with a minima in between. The function grows
exponentially for x>1. This function has an interesting minima that exposes the base 'e'. The minima can be found
using logarithmic differentiation to be at x=1/e. That is a minima at y=(1/e)^{(1/e)}=0.692200628. Look
at the form of the equation: y=x^{x}, similar to y=e^{x}, but charts a neat sort of 'parabolic
looking' curve with a minima involving e located at (x,y) = (1/e,(1/e)^{(1/e)}).

Is this not a surprising relationship?, tying (e), (PI), and (i) into a neat bundle where (i)=(-1)^{½}

Maybe things like integer summations lead to a quotation attributed to mathematician Leopold Kronecker who once wrote that "God made the integers; all else is the work of man". And Einstein said "did God have a choice when creating the universe?" I think that question related to his wonder about the blemish free way that the fabric of the Universe hangs together. See, if one were sitting around designing a Universe, would one remember to tie all these functions and constants and summations together as they are?