§10.25 Definitions

§10.25(i) Modified Bessel’s Equation

 10.25.1 $z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+z\frac{\mathrm{d}w}{\mathrm{d% }z}-(z^{2}+\nu^{2})w=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.1 Referenced by: §10.25(ii), §10.25(iii), §10.27, §10.45, §10.74(ii) Permalink: http://dlmf.nist.gov/10.25.E1 Encodings: TeX, pMML, png See also: Annotations for §10.25(i), §10.25 and Ch.10

This equation is obtained from Bessel’s equation (10.2.1) on replacing $z$ by $\pm iz$, and it has the same kinds of singularities. Its solutions are called modified Bessel functions or Bessel functions of imaginary argument.

§10.25(ii) Standard Solutions

 10.25.2 $I_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}\frac{(\tfrac{1}% {4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}.$ ⓘ Defines: $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.10 Referenced by: §10.30(i), §10.31, §10.38, §10.45, §10.46, §10.53, §8.6(i) Permalink: http://dlmf.nist.gov/10.25.E2 Encodings: TeX, pMML, png See also: Annotations for §10.25(ii), §10.25 and Ch.10

This solution has properties analogous to those of $J_{\nu}\left(z\right)$, defined in §10.2(ii). In particular, the principal branch of $I_{\nu}\left(z\right)$ is defined in a similar way: it corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$.

The defining property of the second standard solution $K_{\nu}\left(z\right)$ of (10.25.1) is

 10.25.3 $K_{\nu}\left(z\right)\sim\sqrt{\pi/(2z)}e^{-z},$ ⓘ Defines: $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\nu$: complex parameter Referenced by: §10.25(ii), §10.25(iii), §10.30(ii) Permalink: http://dlmf.nist.gov/10.25.E3 Encodings: TeX, pMML, png See also: Annotations for §10.25(ii), §10.25 and Ch.10

as $z\to\infty$ in $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta$ $(<\tfrac{3}{2}\pi)$. It has a branch point at $z=0$ for all $\nu\in\mathbb{C}$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$.

Both $I_{\nu}\left(z\right)$ and $K_{\nu}\left(z\right)$ are real when $\nu$ is real and $\operatorname{ph}z=0$.

For fixed $z$ $(\neq 0)$ each branch of $I_{\nu}\left(z\right)$ and $K_{\nu}\left(z\right)$ is entire in $\nu$.

Branch Conventions

Except where indicated otherwise it is assumed throughout the DLMF that the symbols $I_{\nu}\left(z\right)$ and $K_{\nu}\left(z\right)$ denote the principal values of these functions.

Symbol $\mathscr{Z}_{\nu}\left(z\right)$

Corresponding to the symbol $\mathscr{C}_{\nu}$ introduced in §10.2(ii), we sometimes use $\mathscr{Z}_{\nu}\left(z\right)$ to denote $I_{\nu}\left(z\right)$, $e^{\nu\pi i}K_{\nu}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$.

§10.25(iii) Numerically Satisfactory Pairs of Solutions

Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). It is assumed that $\Re\nu\geq 0$. When $\Re\nu<0$, $I_{\nu}\left(z\right)$ is replaced by $I_{-\nu}\left(z\right)$.