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10 Bessel FunctionsModified Bessel Functions

§10.45 Functions of Imaginary Order

With z=x, and ν replaced by iν, the modified Bessel’s equation (10.25.1) becomes

10.45.1 x2d2wdx2+xdwdx+(ν2-x2)w=0.

For ν and x (0,) define

10.45.2 I~ν(x) =(Iiν(x)), K~ν(x) =Kiν(x).
10.45.3 I~-ν(x) =I~ν(x), K~-ν(x) =K~ν(x),

and I~ν(x), K~ν(x) are real and linearly independent solutions of (10.45.1):

10.45.4 𝒲{K~ν(x),I~ν(x)}=1/x.

As x+

10.45.5 I~ν(x) =(2πx)-12ex(1+O(x-1)),
K~ν(x) =(π/(2x))12e-x(1+O(x-1)).

As x0+

10.45.6 I~ν(x)=(sinh(πν)πν)12cos(νln(12x)-γν)+O(x2),

where γν is as in §10.24. The corresponding result for K~ν(x) is given by

10.45.7 K~ν(x)=-(πνsinh(πν))12sin(νln(12x)-γν)+O(x2),

when ν>0, and

10.45.8 K~0(x)=K0(x)=-ln(12x)-γ+O(x2lnx),

where γ again denotes Euler’s constant (§5.2(ii)).

In consequence of (10.45.5)–(10.45.7), I~ν(x) and K~ν(x) comprise a numerically satisfactory pair of solutions of (10.45.1) when x is large, and either I~ν(x) and (1/π)sinh(πν)K~ν(x), or I~ν(x) and K~ν(x), comprise a numerically satisfactory pair when x is small, depending whether ν0 or ν=0.

For graphs of I~ν(x) and K~ν(x) see §10.26(iii).

For properties of I~ν(x) and K~ν(x), including uniform asymptotic expansions for large ν and zeros, see Dunster (1990a). In this reference I~ν(x) is denoted by (1/π)sinh(πν)Liν(x). See also Gil et al. (2003a) and Balogh (1967).