Equation of the Normal Curve

#1
The equation of the normal curve contains the irrational numbers pi and e and looks like a messy equation to the non mathematical person. My question is, how was this equation developed and found to be what it is today, and why is it that pi and e fit so nicely into the equation?
 

spunky

Smelly poop man with doo doo pants.
#3
Math. \(\hspace{.1cm}\)
+1 internetz for this informative post

but, OK, in all seriousness.... if you're unfamiliar with the normal probability density function maybe start up reading on the Gaussian integral and overall properties of \(f(x)=e^{-x^{2}}\)