This falls short of a proof about elementary differential definition of

gamma function, but perhaps it will give insight as to why.

For z not 0 or a negative integer,

gamma(z) = integral(from 0 to infinity)(t^(z-1) e^(-t) dt)

= limit(m-->infinity)(evaluate between 0 and m)

((z-1)t^(z-2) - (z-1)(z-2)/2 t^(z-3) +((z-1)(z-2)(z-3)/3!)t^(z-4)

+ (z!/((z-5)! 4!))t^(z-5) - (z!/((z-6)! 5!)) t^(z-6) + ...)e^(-t)

= limit(m--> infinity)

((z-1)(-m)^(z-2) - (z-1)(z-2)/2 (-m)^(z-3)

+((z-1)(z-2)(z-3)/3!)(-m)^(z-4)

+ (z!/((z-5)! 4!))(-m)^(z-5)

- (z!/((z-6)! 5!)) (-m)^(z-6) + ...) * e^(-m)

Kermit