HP 48gII User's Manual
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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
)( dxex
xα
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
xn
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[)( xx Γ=
ψ
.