§10.1 Special Notation

Keywords:: Bessel functions, Bessel transform, Hankel functions, Kelvin functions, Weber’s function, cylinder functions, derivatives, differential equations, integrals, modified Bessel functions, spherical Bessel functions
Referenced by:: §10.75(ix)
Permalink:: http://dlmf.nist.gov/10.1

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

$m,n$	integers. In §§10.47–10.71 $n$ is nonnegative.
$k$	nonnegative integer (except in §10.73).
$x,y$	real variables.
$z$	complex variable.
$\nu$	real or complex parameter (the order).
$\delta$	arbitrary small positive constant.
$\vartheta$	$z(\!\ifrac{d}{dz})$ .
$\mathop{\psi\/}\nolimits\!\left(x\right)$	${\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(x\right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$ : logarithmic derivative of the gamma function (§5.2(i)).
primes	derivatives with respect to argument, except where indicated otherwise.

The main functions treated in this chapter are the Bessel functions $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ ; Hankel functions $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ , $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ ; modified Bessel functions $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ ; spherical Bessel functions $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ , $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ , $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ ; modified spherical Bessel functions $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ , $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ ; Kelvin functions $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits\!\left(x\right)$ . For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$ , it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real.

A common alternative notation for $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ is $N_{{\nu}}(z)$ . Other notations that have been used are as follows.

Abramowitz and Stegun (1964): $j_{n}(z)$ , $y_{n}(z)$ , $h_{n}^{{(1)}}(z)$ , $h_{n}^{{(2)}}(z)$ , for $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ , $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ , $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ , respectively, when $n\geq 0$ .

Jeffreys and Jeffreys (1956): $\mathrm{Hs}_{\nu}(z)$ for $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ , $\mathrm{Hi}_{\nu}(z)$ for $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ , $\mathrm{Kh}_{\nu}(z)$ for $(2/\pi)\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ .

Whittaker and Watson (1927): $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ for $\mathop{\cos\/}\nolimits(\nu\pi)\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ .

For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).