10.44 Sums10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

§10.45 Functions of Imaginary Order

With z=x, and \nu replaced by i\nu, the modified Bessel’s equation (10.25.1) becomes

10.45.1 x^{2}\frac{{d}^{2}w}{{dx}^{2}}+x\frac{dw}{dx}+(\nu^{2}-x^{2})w=0.

For \nu\in\Real and x \in(0,\infty) define

10.45.2 \displaystyle \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right) =\realpart{(\mathop{I_{{i\nu}}\/}\nolimits\!\left(x\right))},\displaystyle \mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right) =\mathop{K_{{i\nu}}\/}\nolimits\!\left(x\right).

Then

10.45.3 \displaystyle \mathop{\widetilde{I}_{{-\nu}}\/}\nolimits\!\left(x\right) =\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right),\displaystyle \mathop{\widetilde{K}_{{-\nu}}\/}\nolimits\!\left(x\right) =\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right),

and \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right), \mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right) are real and linearly independent solutions of (10.45.1):

10.45.4 \mathop{\mathscr{W}\/}\nolimits\{\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right),\mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right)\}=1/x.

In consequence of (10.45.5)–(10.45.7), \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right) and \mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right) comprise a numerically satisfactory pair of solutions of (10.45.1) when x is large, and either \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right) and (1/\pi)\mathop{\sinh\/}\nolimits(\pi\nu)\mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right), or \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right) and \mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right), comprise a numerically satisfactory pair when x is small, depending whether \nu\neq 0 or \nu=0.

For graphs of \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right) and \mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right) see §10.26(iii).

For properties of \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right) and \mathop{\widetilde{K}_{{\nu}}\/}\nolimits\!\left(x\right), including uniform asymptotic expansions for large \nu and zeros, see Dunster (1990a). In this reference \mathop{\widetilde{I}_{{\nu}}\/}\nolimits\!\left(x\right) is denoted by (1/\pi)\mathop{\sinh\/}\nolimits(\pi\nu)L_{{i\nu}}(x). See also Gil et al. (2003a) and Balogh (1967).