##Gamma(pi+1) ~~ 7.188082729##
Strictly speaking, factorial is only defined for non-negative integers.
The usual recursive definition is:
##{ (0! = 1), (n! = n(n-1)! ” for ” n > 0) :}##
The normal way to extend the definition beyond non-negative integers is the Gamma function ##Gamma(x)##, which satisfies:
##Gamma(n) = (n-1)!##
for all positive integer values of ##n##
So using the Gamma function, “pi factorial” is ##Gamma(pi+1)##
For positive Real numbers (and Complex numbers with a positive Real part) we can define:
##Gamma(t) = int_(x=0)^oo x^(t-1) e^(-x) dx##
It is then possible to extend the definition to all Complex numbers, except the negative integers.
With this definition we find:
##Gamma(pi+1) ~~ 7.188082729##
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