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