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10 Bessel FunctionsBessel and Hankel Functions

§10.24 Functions of Imaginary Order

With z=x and ν replaced by iν, Bessel’s equation (10.2.1) becomes

10.24.1 x2d2wdx2+xdwdx+(x2+ν2)w=0.

For ν and x (0,) define

10.24.2 J~ν(x) =sech(12πν)(Jiν(x)),
Y~ν(x) =sech(12πν)(Yiν(x)),


10.24.3 Γ(1+iν)=(πνsinh(πν))12eiγν,

where γν is real and continuous with γ0=0; compare (5.4.3). Then

10.24.4 J~ν(x) =J~ν(x),
Y~ν(x) =Y~ν(x),

and J~ν(x), Y~ν(x) are linearly independent solutions of (10.24.1):

10.24.5 𝒲{J~ν(x),Y~ν(x)}=2/(πx).

As x+, with ν fixed,

10.24.6 J~ν(x) =2/(πx)cos(x14π)+O(x32),
Y~ν(x) =2/(πx)sin(x14π)+O(x32).

As x0+, with ν fixed,

10.24.7 J~ν(x) =(2tanh(12πν)πν)12cos(νln(12x)γν)+O(x2),
10.24.8 Y~ν(x) =(2coth(12πν)πν)12sin(νln(12x)γν)+O(x2),


10.24.9 Y~0(x)=Y0(x)=2π(ln(12x)+γ)+O(x2lnx),

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

In consequence of (10.24.6), when x is large J~ν(x) and Y~ν(x) comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when x is small either J~ν(x) and tanh(12πν)Y~ν(x) or J~ν(x) and Y~ν(x) comprise a numerically satisfactory pair depending whether ν0 or ν=0.

For graphs of J~ν(x) and Y~ν(x) see §10.3(iii).

For mathematical properties and applications of J~ν(x) and Y~ν(x), including zeros and uniform asymptotic expansions for large ν, see Dunster (1990a). In this reference J~ν(x) and Y~ν(x) are denoted respectively by Fiν(x) and Giν(x).