The gamma distribution generalizes the Erlang distribution by allowing " k " to be any real number, using the gamma function instead of the factorial function.
32.
But, just to keep the convoluting complex analysis stuff out, let's consider the factorial function x ! extended to all real numbers save for negative integers.
33.
So if you have a function involving the factorial function, eg f ( x ) = x ! * x ^ n, how do you go about differentiating it?
34.
For example, the factorial function may be implemented iteratively in C by assigning to an loop index variable and accumulator variable, rather than passing arguments and returning values by recursion:
35.
The first line of the factorial function describes the " type " of this function; while it is optional, it is considered to be good style to include it.
36.
The definition of the factorial function can also be extended to non-integer arguments, while retaining its most important properties; this involves more advanced mathematics, notably techniques from mathematical analysis.
37.
Using the product function from the Prelude, a number of small functions analogous to standard library, and using the Haskell syntax for arithmetic sequences, the factorial function can be expressed in Haskell as follows:
38.
Like tetration, there is currently no accepted method of extension of the exponential factorial function to complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function.
39.
Like tetration, there is currently no accepted method of extension of the exponential factorial function to complex values of its argument, unlike the factorial function, for which such an extension is provided by the gamma function.
40.
To achieve this De Moivre developed an asymptotic sequence for the factorial function -which we now refer to as Stirling's approximation -and Bernoulli's formula for the sum of powers of numbers.
