 In 1840 Joseph Ludwig Raabe proved that
In particular, if formula_197 then

The latter can be derived taking the logarithm in the above multiplication formula, which gives an expression for the Riemann sum of the integrand.Taking the limit for formula_199 gives the formula.An alternative notation which was originally introduced by Gauss and which was sometimes used is the formula_200-function, which in terms of the gamma function is

so that formula_202 for every non-negative integer formula_33.