5 Gamma Function Notation 5 Gamma Function 5.2 Definitions

§5.1 Special Notation

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Keywords:: digamma function, double gamma function, gamma function, notation, psi function
Permalink:: http://dlmf.nist.gov/5.1
See also:: Annotations for Ch.5

(For other notation see Notation for the Special Functions.)

$j, m, n$	nonnegative integers.
$k$	nonnegative integer, except in §5.20.
$x, y$	real variables.
$z=x+iy$	complex variable.
$a, b, q, s, w$	real or complex variables with $\|q\|<1$ .
$\delta$	arbitrary small positive constant.
$\gamma$	Euler’s constant (§5.2(ii)).
primes	derivatives with respect to the variable.

The main functions treated in this chapter are the gamma function $\Gamma\left(z\right)$ , the psi function (or digamma function) $\psi\left(z\right)$ , the beta function $\mathrm{B}\left(a,b\right)$ , and the $q$ -gamma function $\Gamma_{q}\left(z\right)$ .

The notation $\Gamma\left(z\right)$ is due to Legendre. Alternative notations for this function are: $\Pi(z-1)$ (Gauss) and $(z-1)!$ . Alternative notations for the psi function are: $\Psi(z-1)$ (Gauss) Jahnke and Emde (1945); $\Psi(z)$ Davis (1933); $\mathsf{F}(z-1)$ Pairman (1919).

5 Gamma Function 5.2 Definitions

NIST

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