# §10.1 Special Notation

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

 integers. In §§10.47–10.71 is nonnegative. nonnegative integer (except in §10.73). real variables. complex variable. real or complex parameter (the order). arbitrary small positive constant. . : 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 , ; Hankel functions , ; modified Bessel functions , ; spherical Bessel functions , , , ; modified spherical Bessel functions , , ; Kelvin functions , , , . For the spherical Bessel functions and modified spherical Bessel functions the order is a nonnegative integer. For the other functions when the order is replaced by , it can be any integer. For the Kelvin functions the order is always assumed to be real.

A common alternative notation for is . Other notations that have been used are as follows.

Abramowitz and Stegun (1964): , , , , for , , , , respectively, when .

Jeffreys and Jeffreys (1956): for , for , for .

Whittaker and Watson (1927): for .

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