# §10.1 Special Notation

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

$m,n$ integers. In §§10.47–10.71 $n$ is nonnegative. nonnegative integer (except in §10.73). real variables. complex variable. real or complex parameter (the order). arbitrary small positive constant. $z(\!\ifrac{\mathrm{d}}{\mathrm{d}z})$. $\Gamma'\left(x\right)/\Gamma\left(x\right)$: logarithmic derivative of the gamma function (§5.2(i)). 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).