TRIGAMMA

TRIGAMMA (LET)

Library Function

Compute the trigamma function.

The digamma function is the logarithmic derivative of the gamma function and is defined as:

\[ \psi(x) = \frac{\Gamma'(x)} {\Gamma(x)} \]

where \( \Gamma \) is the gamma function and \( \Gamma' \) is the derivative of the gamma function.

The trigamma function is the derivative of the digamma function and is defined as

\[ \begin{array}{lcl} \psi_{1}(x) & = & \psi'(x) \\ & = & \frac{d^2}{dx^2} \ln (\Gamma(x)) \\ & = & \sum_{n=0}^{\infty}{\frac{1}{(x+n)^{2}}} \end{array} \]

This function is defined for positive numbers.

LET <y> = TRIGAMMA(<x>)             <SUBSET/EXCEPT/FOR qualification>
where <x> is a number, variable or a parameter;
            <y> is a variable or a parameter (depending on what <x> is) where the computed trigamma values are stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = TRIGAMMA(1)
LET X2 = TRIGAMMA(X1)
LET X2 = TRIGAMMA(X1-4)

Dataplot uses the routine DPSIFN from the SLATEC Common Mathematical Library to compute this function. SLATEC is a large set of high quality, portable, public domain Fortran routines for various mathematical capabilities maintained by seven federal laboratories.

None

None

GAMMA	=	Compute the gamma function.
DIGAMMA	=	Compute the digamma function.
LOGGAMMA	=	Compute the log (to base e) gamma function.

D. E. Amos (1983), "A portable Fortran subroutine for derivatives of the Psi function", Algorithm 610, ACM Transactions on Mathematical Software 9, 4, pp. 494-502.

Abramowitz and Stegun, "Handbook of Mathematical Functions, Applied Mathematics Series, Vol. 55", National Bureau of Standards, 1964 (chapter 6).

Special Functions

2014/12

 
TITLE CASE ASIS
TITLE Trigamma Function
PLOT TRIGAMMA(X) FOR X = 0.01 0.01 10