10 Bessel Functions Modified Bessel Functions 10.24 Functions of Imaginary Order 10.26 Graphics

§10.25 Definitions

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Referenced by:: §2.8(i)
Permalink:: http://dlmf.nist.gov/10.25
See also:: Annotations for Ch.10

§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

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Keywords:: definitions, differential equations, modified Bessel functions, modified Bessel’s equation, singularities
Permalink:: http://dlmf.nist.gov/10.25.i
See also:: Annotations for §10.25 and Ch.10

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

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : 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 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

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Keywords:: modified Bessel functions, modified Bessel’s equation, principal branches (or values), standard solutions
Notes:: See Olver (1997b, pp. 60, 236–237, 250).
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
See also:: Annotations for §10.25 and Ch.10

10.25.2 $I_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}\frac{(\tfrac{1}% {4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}.$

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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 ), : nonnegative integer, : 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 $K_{\nu}\left(z\right)\sim\sqrt{\pi/(2z)}e^{-z},$

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

Branch Conventions

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Keywords:: branch conventions, modified Bessel functions
See also:: Annotations for §10.25(ii), §10.25 and Ch.10

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

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Defines:: $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified cylinder function
See also:: Annotations for §10.25(ii), §10.25 and Ch.10

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 and $\nu$ .

§10.25(iii) Numerically Satisfactory Pairs of Solutions

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Keywords:: modified Bessel’s equation, numerically satisfactory 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).
Referenced by:: §14.20(i)
Permalink:: http://dlmf.nist.gov/10.25.iii
See also:: Annotations for §10.25 and Ch.10

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 $I_{-\nu}\left(z\right)$ .

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

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