§5.1 Special Notation

Keywords:: digamma function, double gamma function, gamma function, psi function
Permalink:: http://dlmf.nist.gov/5.1

(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.
$\EulerConstant$	Euler’s constant (§5.2(ii)).
primes	derivatives with respect to the variable.

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

The notation $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ 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).