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Subject: An Explicit Formula for Factorial
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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.)