With , and replaced by , the modified Bessel’s equation (10.25.1) becomes
For and define
and , are real and linearly independent solutions of (10.45.1):
where is as in §10.24. The corresponding result for is given by
when , and
where again denotes Euler’s constant (§5.2(ii)).
In consequence of (10.45.5)–(10.45.7), and comprise a numerically satisfactory pair of solutions of (10.45.1) when is large, and either and , or and , comprise a numerically satisfactory pair when is small, depending whether or .
For graphs of and see §10.26(iii).