HP 49g+ User Manual page 137

The Gamma function Γ(α)
GAMMA:
PSI: N-th derivative of the digamma function
Psi: Digamma function, derivative of the ln(Gamma)
The Gamma function is defined by
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,
n
d
( , ) ψ (
i.e., n x
n
dx
Psi function. For this function, y must be a positive integer.
The Psi function, ψ(x), or digamma function, is defined as
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:
∞
∫
( α )
0
)
, where ψ(x) is known as the digamma function, or
x
α − 1 − x
x e dx . This function has
ψ ( x ) ln[ (
Page 3-15
x )]
.