------------------------------------------------------------------------ Subject: An Explicit Formula for Factorial ------------------------------------------------------------------------ I tried to convince you that recursively defined functions were "nicer" (e.g., easier to understand and to compute with) than "explicitly" defined functions. The main example I gave you was the Fibonacci function, which is pretty ugly in its explicit form, with all those square roots of 5 raised to the nth power, but is very neat and clean in its more familiar recursive definition. I also showed you the "Big Pi" notation that can be used to define the factorial function, but I suggested that even that was not completely explicit. There's another explicit definition of n! that is even uglier; you can see it at: http://en.wikipedia.org/wiki/Gamma_function For those of you who might not want to read the entire Wikipedia article, GAMMA is a function over complex numbers. Briefly, GAMMA(z) =def INTEGRAL from 0 to infinity of (t^(z-1))*(e^-t)) dt (You don't have to understand that; you just have to realize that it's not as nice as the recursive definition of factorial.) It turns out that GAMMA(n+1) = n! (It also turns out that (GAMMA(1/2))^2 = pi, which I think is pretty amazing, too.)