HP 50G, Руководство пользователя

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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 . This function has
applications in applied mathematics for science and engineering, as well as in
probability and statistics.
The PSI function,
Ψ
(x,y) , represents the y-th derivative of the digamma function,
i.e., , 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 .
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.
∫
∞
− −
= Γ
0
1
) (
dx e x
x
α
α
) ( ) , (
x
dx
d
x n
n
n
ψ
= Ψ
)] ( ln[ ) ( x x Γ =
ψ