# §10.45 Functions of Imaginary Order

With , and replaced by , the modified Bessel’s equation (10.25.1) becomes

For and define

10.45.2

Then

and , are real and linearly independent solutions of (10.45.1):

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).

For properties of and , including uniform asymptotic expansions for large and zeros, see Dunster (1990a). In this reference is denoted by . See also Gil et al. (2003a) and Balogh (1967).