HP 49g+ - Manual (Page 137)

Page 3-15
GAMMA: The Gamma function Γ(α)
PSI: N-th derivative of the digamma function
Psi: Digamma function, derivative of the ln(Gamma)
The Gamma function is defined by
∫
∞
− − = Γ
0
1 ) (
dx e x
x α α
. This function has
applications in applied mathematics for science and engineering, as well as
in probability and statistics.
Factorial of a number
The factorial of a positive integer number n is defined as n!=n ⋅(n-1)⋅(n-
2) …3 ⋅2⋅1, with 0! = 1. The factorial function is available in the calculator by
using ~‚2. In both ALG and RPN modes, enter the number first,
followed by the sequence ~‚2. Example: 5~‚2`.
The Gamma function, defined above, has the property that
Γ(α) = (α−1) Γ(α−1), for α > 1.
Therefore, it can be related to the factorial of a number, i.e., Γ(α) = (α−1)!,
when α is a positive integer. We can also use the factorial function to
calculate the Gamma function, and vice versa. For example, Γ(5) = 4! or,
4~‚2`. The factorial function is available in the MTH menu,
through the 7. PROBABILITY.. menu.
The PSI function,
Ψ(x,y), represents the y-th derivative of the digamma function,
i.e.,
) ( ) , (
x
dx
d
x n
n
n
ψ = Ψ
, where ψ(x) is known as the digamma function, or
Psi function. For this function, y must be a positive integer.
The Psi function,
ψ(x), or digamma function, is defined as )] ( ln[ ) ( x x Γ = ψ
.
Examples of these special functions are shown here using both the ALG and
RPN modes. As an exercise, verify that GAMMA(2.3) = 1.166711…,
PSI(1.5,3) = 1.40909.., and Psi(1.5) = 3.64899739..E-2.
These calculations are shown in the following screen shot: