# §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. $z(\!\ifrac{d}{dz})$. ${\mathop{\Gamma\/}\nolimits^{\prime}}\!\left(x\right)/\mathop{\Gamma\/}% \nolimits\!\left(x\right)$: 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 $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$, $\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)$; 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).