**bs@faron.UUCP (Robert D. Silverman)** (04/02/85)

In response to nonsense that has been coming across the net regarding the derivative of the factorial function: The factorial function is defined only for the non-negative integers (0! = 1) and thus is infinitely discontinuous and NOWHERE differentiable. Talking about its derivative is pure nonsense. It does have a continuous analog known as the gamma function and I present some facts about it here. Some facts about the gamma function.... The gamma function is an analytic function except at 0, -1, -2, -3, ... where it has simple poles. It can be represented as an infinite product in Weirstrauss canonical form as: Letting T(z) be the gamma function and y be the Euler-Mascheroni constant inf 1 --- --- = z exp(yz) | | { (1+z/n) exp (-z/n) } T(z) --- n = 1 The gamma function satisfies no differential equation with rational coefficients and may be represented in a variety of forms. It does, however satisfy the difference equation T(z) = zT(z-1) from which we get the the factorial function for positive integers. It may also be represented as: inf 1 --- z -1 T(z) = -- | | { (1 + 1/n) (1 + z/n) } z --- n = 1 except at the non-positive integers. It satisfies: (known as the duplication formula) 2z-1 2 T(z) T(z + 1/2) = sqrt(PI) T(2z) It is relatively easy to find a formula for the logarithmic derivative of the gamma function: inf / -t -zt d T(z) | e e ------ = | { --- - --------- } dt d z | t (1 - exp(-t)) / 0 . The formula for the derivative itself is rather more difficult: inf -- __ / | d T(z) | 1 | t dt | ------ = T(z) | log(z) - --- - 2 | --------------------- | d z | 2z | 2 2 | -- / (t + z )exp(2PIt - 1) | 0 -- It can be derived from Binet's formula for the log(T(z)). The gamma function itself is usually introduced (defined as) an infinite integral: inf / | | -t z-1 T(z) = | e t dt | / 0 However, this integral converges only for REAL(z) > 0 , and thus fails to be an analytic representation over the entire plane. For many more relations involving the gamma function see: "Handbook of Mathematical Functions" , Abramowitz and Stegun eds. National Bureau of Standards