§10.25 Definitions

Referenced by:: §2.8(i)
Permalink:: http://dlmf.nist.gov/10.25

§10.25(i) Modified Bessel’s Equation
§10.25(ii) Standard Solutions
§10.25(iii) Numerically Satisfactory Pairs of Solutions

§10.25(i) Modified Bessel’s Equation

Keywords:: differential equations, modified Bessel functions, modified Bessel’s equation, singularities
Permalink:: http://dlmf.nist.gov/10.25.i

10.25.1 $z^{2}\frac{{d}^{2}w}{{dz}^{2}}+z\frac{dw}{dz}-(z^{2}+\nu^{2})w=0.$

Symbols:: $\frac{df}{dx}$ : 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
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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

Notes:: See Olver (1997b, pp. 60, 236–237, 250).
Keywords:: modified Bessel functions, modified Bessel’s equation
Referenced by:: §10.21(iv), ¶ ‣ §10.22(ii), ¶ ‣ §10.22(iv), §10.22(iv), §10.29(i), §10.43(i), ¶ ‣ §10.9(iii), §10.9(iii), §11.1, §12.7(iii), §13.10(ii), §13.16(i), §13.18(iii), §13.21(i), §13.23(i), §13.24(ii), §13.4(i), §13.6(iii), §13.8(iii), §14.15(i), §14.20(vii), §15.12(iii), §18.12, ¶ ‣ §18.18(vii), §2.8(iv), §26.10(vi), §28.24, §33.20(i), §33.20(ii), §33.20(iii), §33.9(ii), §6.7(iii), §8.6(i), §9.12(vii), §9.13(i), §9.6(i), §9.6(ii)
Permalink:: http://dlmf.nist.gov/10.25.ii

10.25.2 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum _{{k=0}}^{\infty}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!\mathop{\Gamma\/}\nolimits\!\left(\nu+k+1\right)}.$

Defines:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $!$ : $n!$ : factorial, $k$ : 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
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This solution has properties analogous to those of $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ , defined in §10.2(ii). In particular, the principal branch of $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ is defined in a similar way: it corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$ , is analytic in $\Complex\setminus(-\infty,0]$ , and two-valued and discontinuous on the cut $\mathop{\mathrm{ph}\/}\nolimits z=\pm\pi$ .

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

10.25.3 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)\sim\sqrt{\pi/(2z)}e^{{-z}},$

Defines:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function
Symbols:: $e$ : base of exponential function, $\sim$ : asymptotic equality, $z$ : 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
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as $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{2}\pi-\delta$ $(<\tfrac{3}{2}\pi)$ . It has a branch point at $z=0$ for all $\nu\in\Complex$ . The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\Complex\setminus(-\infty,0]$ , and two-valued and discontinuous on the cut $\mathop{\mathrm{ph}\/}\nolimits z=\pm\pi$ .

Both $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ and $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ are real when $\nu$ is real and $\mathop{\mathrm{ph}\/}\nolimits z=0$ .

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

¶ Branch Conventions

Keywords:: modified Bessel functions

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

¶ Symbol $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$

Defines:: $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function

Corresponding to the symbol $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits$ introduced in §10.2(ii), we sometimes use $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ to denote $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ , $e^{{\nu\pi i}}\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ , 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

Notes:: The conclusions in this subsection follow from §2.7(iv) and the limiting forms of the solutions as $z\to 0$ and $z\to\infty$ ; see (10.25.3) and §10.30. See also (10.27.3).
Keywords:: modified Bessel’s equation
Referenced by:: §14.20(i)
Permalink:: http://dlmf.nist.gov/10.25.iii

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

Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.

Pair	Region
$\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right),\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$	$\|\mathop{\mathrm{ph}\/}\nolimits z\|\leq\tfrac{1}{2}\pi$
$\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right),\mathop{K_{{\nu}}\/}\nolimits\!\left(ze^{{\mp\pi i}}\right)$	$\tfrac{1}{2}\pi\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2}\pi$

Symbols:: $e$ : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.25(iii)
Permalink:: http://dlmf.nist.gov/10.25.T1