A related discussion on interpolation: factorial and gamma-function

[Gottfried]

It says "If n = 0,-1,-2,... then Gamma(n) becomes infinite.". Gamma of zero or a negative integer is undefined, not infinite. For example, . One divided by zero is not infinite, it is undefined.

Peter Luschny has once explained this to me with the (standard) concept of limiting towards an infinitesimal interval. See the approximation of the quotient in \( \lim_{h \to 0} { \Gamma (0 \pm h) \over \zeta(1 \pm h) } \) :

Code:

   .
h=0.001;[gamma(0+h)/zeta(1+h),gamma(0-h)/zeta(1-h)]
%9 = [0.998847149536, 1.00115601566]

h=1e-5;[gamma(0+h)/zeta(1+h),gamma(0-h)/zeta(1-h)]
%11 = [0.999988455845, 1.00001154447]

h=1e-6;[gamma(0+h)/zeta(1+h),gamma(0-h)/zeta(1-h)]
%13 = [0.999998845570, 1.00000115443]

h=1e-12;[gamma(0+h)/zeta(1+h),gamma(0-h)/zeta(1-h)]
%15 = [0.999999999999, 1.00000000000]

...