10 Bessel Functions Notation 10 Bessel Functions 10.2 Definitions

§10.1 Special Notation

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Keywords:: Bessel functions, Bessel transform, Hankel functions, Kelvin functions, Weber’s function, asymptotic expansions for large order, cylinder functions, derivatives, differential equations, integrals, modified Bessel functions, notation, of imaginary argument, spherical Bessel functions, zeros
Referenced by:: 1st item
Permalink:: http://dlmf.nist.gov/10.1
See also:: Annotations for Ch.10

(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.
$\vartheta$	$z(\!\ifrac{\mathrm{d}}{\mathrm{d}z})$ .
$\psi\left(x\right)$	$\Gamma'\left(x\right)/\Gamma\left(x\right)$ : 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 $J_{\nu}\left(z\right)$ , $Y_{\nu}\left(z\right)$ ; Hankel functions ${H^{(1)}_{\nu}}\left(z\right)$ , ${H^{(2)}_{\nu}}\left(z\right)$ ; modified Bessel functions $I_{\nu}\left(z\right)$ , $K_{\nu}\left(z\right)$ ; spherical Bessel functions $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , ${\mathsf{h}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{h}^{(2)}_{n}}\left(z\right)$ ; modified spherical Bessel functions ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{i}^{(2)}_{n}}\left(z\right)$ , $\mathsf{k}_{n}\left(z\right)$ ; Kelvin functions $\operatorname{ber}_{\nu}\left(x\right)$ , $\operatorname{bei}_{\nu}\left(x\right)$ , $\operatorname{ker}_{\nu}\left(x\right)$ , $\operatorname{kei}_{\nu}\left(x\right)$ . 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 $Y_{\nu}\left(z\right)$ 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 $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , ${\mathsf{h}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{h}^{(2)}_{n}}\left(z\right)$ , respectively, when $n\geq 0$ .

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

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

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

10 Bessel Functions 10.2 Definitions

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