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