Throughout this section is real and nonpositive.
Graphs of and are included in §9.3(i). The branches of and are continuous and fixed by . (These definitions of and differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).)
In terms of Bessel functions, and with ,
Primes denote differentiations with respect to , which is continued to be assumed real and nonpositive.
As increases from to 0 each of the functions , , , , , is increasing, and each of the functions , , is decreasing.