10 Bessel Functions Bessel and Hankel Functions 10.1 Special Notation 10.3 Graphics

§10.2 Definitions

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Keywords:: Bessel functions, definitions
Referenced by:: §10.74(i), §2.8(i), §32.10(iii)
Permalink:: http://dlmf.nist.gov/10.2
See also:: Annotations for Ch.10

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

§10.2(i) Bessel’s Equation

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Keywords:: Bessel functions, Bessel’s equation, Hankel functions, cylinder functions, differential equations, singularities
Permalink:: http://dlmf.nist.gov/10.2.i
See also:: Annotations for §10.2 and Ch.10

10.2.1 $z^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+z\frac{\mathrm{d}w}{\mathrm{d% }z}+(z^{2}-\nu^{2})w=0.$

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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:: 9.1.1
Referenced by:: §10.13, §10.15, §10.2(ii), §10.2(ii), §10.2(iii), §10.24, §10.25(i), §10.27, §10.4, §10.61(ii), §10.74(ii)
Permalink:: http://dlmf.nist.gov/10.2.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.2(i), §10.2 and Ch.10

This differential equation has a regular singularity at with indices $\pm\nu$ , and an irregular singularity at $z=\infty$ of rank 1; compare §§2.7(i) and 2.7(ii).

§10.2(ii) Standard Solutions

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Keywords:: Bessel functions, Bessel’s equation, of the first, second, and third kinds, principal branches (or values), standard solutions
Notes:: See Olver (1997b, pp. 57, 237–238, 242–243) and Watson (1944, pp. 38–45, 57–64, 196–198).
Referenced by:: §1.17(ii), §1.18(vi), §10.14, §10.22(i), §10.25(ii), §10.25(ii), §10.6(i), §11.1, §12.14(vii), §13.10(v), §13.16(i), §13.21(i), §13.21(ii), §13.23(iii), §13.24(ii), §13.4(i), §13.8(iii), §14.15(iii), §18.10(iv), §18.11(ii), §18.12, §18.15(iv), §18.15(i), §18.17(v), §2.5(i), §2.8(iv), §28.20(iii), §28.23, §3.11(iii), §3.5(iii), §3.5(viii), §30.11(i), §31.10(ii), §33.20(i), §33.20(ii), §33.20(iii), §33.5(ii), §33.9(ii), §36.2(ii), §8.6(i), §9.11(iii), §9.13(i), §9.6(i), §9.6(ii), Notation for the Special Functions
Permalink:: http://dlmf.nist.gov/10.2.ii
See also:: Annotations for §10.2 and Ch.10

Bessel Function of the First Kind

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Keywords:: Bessel functions, of the first kind
See also:: Annotations for §10.2(ii), §10.2 and Ch.10

10.2.2 $J_{\nu}\left(z\right)=(\tfrac{1}{2}z)^{\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{(% \tfrac{1}{4}z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)}.$

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Defines:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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:: 9.1.10
Referenced by:: §10.11, §10.16, §10.19(i), §10.22(ii), §10.22(iii), §10.24, §10.53, §10.65(i), §10.7(i), §10.74(ii), §10.8, §18.38(iii), §8.6(i)
Permalink:: http://dlmf.nist.gov/10.2.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

This solution of (10.2.1) is an analytic function of $z\in\mathbb{C}$ , except for a branch point at when $\nu$ is not an integer. The principal branch of $J_{\nu}\left(z\right)$ corresponds to the principal value of $(\tfrac{1}{2}z)^{\nu}$ (§4.2(iv)) and is analytic in the -plane cut along the interval $(-\infty,0]$ .

When $\nu=n$ $(\in\mathbb{Z})$ , $J_{\nu}\left(z\right)$ is entire in .

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

Bessel Function of the Second Kind (Weber’s Function)

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Keywords:: Bessel functions, of the second kind
See also:: Annotations for §10.2(ii), §10.2 and Ch.10

10.2.3 $Y_{\nu}\left(z\right)=\frac{J_{\nu}\left(z\right)\cos\left(\nu\pi\right)-J_{-% \nu}\left(z\right)}{\sin\left(\nu\pi\right)}.$

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Defines:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind
Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.2
Referenced by:: §10.11, §10.15, §10.19(i), §10.2(ii), §10.22(ii), §10.24, §10.4, §10.47(ii), §10.7(i), §10.8, §11.10(v), §11.4(i)
Permalink:: http://dlmf.nist.gov/10.2.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

When $\nu$ is an integer the right-hand side is replaced by its limiting value:

10.2.4 $Y_{n}\left(z\right)=\frac{1}{\pi}\left.\frac{\partial J_{\nu}\left(z\right)}{% \partial\nu}\right|_{\nu=n}+\left.\frac{(-1)^{n}}{\pi}\frac{\partial J_{\nu}% \left(z\right)}{\partial\nu}\right|_{\nu=-n},$ $n=0,\pm 1,\pm 2,\dotsc$ .

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , : integer, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.15, §10.2(ii), §10.22(iv), §10.4
Permalink:: http://dlmf.nist.gov/10.2.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

Whether or not $\nu$ is an integer $Y_{\nu}\left(z\right)$ has a branch point at . The principal branch corresponds to the principal branches of $J_{\pm\nu}\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the -plane along the interval $(-\infty,0]$ .

Except in the case of $J_{\pm n}\left(z\right)$ , the principal branches of $J_{\nu}\left(z\right)$ and $Y_{\nu}\left(z\right)$ are two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$ ; compare §4.2(i).

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

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

Bessel Functions of the Third Kind (Hankel Functions)

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Keywords:: Bessel functions, Hankel functions, definitions, limiting forms, of the third kind, principal branches (or values)
See also:: Annotations for §10.2(ii), §10.2 and Ch.10

These solutions of (10.2.1) are denoted by ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ , and their defining properties are given by

10.2.5 ${H^{(1)}_{\nu}}\left(z\right)\sim\sqrt{2/(\pi z)}e^{i(z-\frac{1}{2}\nu\pi-% \frac{1}{4}\pi)}$

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Defines:: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function)
Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.2.3
Referenced by:: §10.2(ii), §10.7(ii)
Permalink:: http://dlmf.nist.gov/10.2.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

as $z\to\infty$ in $-\pi+\delta\leq\operatorname{ph}z\leq 2\pi-\delta$ , and

10.2.6 ${H^{(2)}_{\nu}}\left(z\right)\sim\sqrt{2/(\pi z)}e^{-i\left(z-\frac{1}{2}\nu% \pi-\frac{1}{4}\pi\right)}$

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Defines:: ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function)
Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.2.4
Referenced by:: §10.2(ii), §10.7(ii)
Permalink:: http://dlmf.nist.gov/10.2.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

as $z\to\infty$ in $-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta$ , where $\delta$ is an arbitrary small positive constant. Each solution has a branch point at for all $\nu\in\mathbb{C}$ . The principal branches correspond to principal values of the square roots in (10.2.5) and (10.2.6), again with a cut in the -plane along the interval $(-\infty,0]$ .

The principal branches of ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ are two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$ .

For fixed $(\neq 0)$ each branch of ${H^{(1)}_{\nu}}\left(z\right)$ and ${H^{(2)}_{\nu}}\left(z\right)$ is entire in $\nu$ .

Branch Conventions

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Keywords:: Bessel functions, Hankel functions, branch conventions
See also:: Annotations for §10.2(ii), §10.2 and Ch.10

Except where indicated otherwise, it is assumed throughout the DLMF that the symbols $J_{\nu}\left(z\right)$ , $Y_{\nu}\left(z\right)$ , ${H^{(1)}_{\nu}}\left(z\right)$ , and ${H^{(2)}_{\nu}}\left(z\right)$ denote the principal values of these functions.

Cylinder Functions

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Defines:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function
Keywords:: Bessel functions, Bessel’s equation, cylinder functions, definition, definitions, standard solutions
See also:: Annotations for §10.2(ii), §10.2 and Ch.10

The notation $\mathscr{C}_{\nu}\left(z\right)$ denotes $J_{\nu}\left(z\right)$ , $Y_{\nu}\left(z\right)$ , ${H^{(1)}_{\nu}}\left(z\right)$ , ${H^{(2)}_{\nu}}\left(z\right)$ , or any nontrivial linear combination of these functions, the coefficients in which are independent of and $\nu$ .

§10.2(iii) Numerically Satisfactory Pairs of Solutions

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Keywords:: Bessel functions, Bessel’s equation, numerically satisfactory solutions, of the first, second, and third kinds, principal branches (or values)
Notes:: The conclusions in this subsection follow from §2.7(iv) and the limiting forms of the solutions as $z\to 0$ and as $z\to\infty$ ; see §10.7.
Permalink:: http://dlmf.nist.gov/10.2.iii
See also:: Annotations for §10.2 and Ch.10

Table 10.2.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.2.1) for the stated intervals or regions in the case $\Re\nu\geq 0$ . When $\Re\nu<0$ , $\nu$ is replaced by $-\nu$ throughout.

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.

Pair	Interval or Region
$J_{\nu}\left(x\right),Y_{\nu}\left(x\right)$	$0<x<\infty$
$J_{\nu}\left(z\right),Y_{\nu}\left(z\right)$	neighborhood of 0 in $\|\operatorname{ph}z\|\leq\pi$
$J_{\nu}\left(z\right),{H^{(1)}_{\nu}}\left(z\right)$	$0\leq\operatorname{ph}z\leq\pi$
$J_{\nu}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)$	$-\pi\leq\operatorname{ph}z\leq 0$
${H^{(1)}_{\nu}}\left(z\right),{H^{(2)}_{\nu}}\left(z\right)$	neighborhood of $\infty$ in $\|\operatorname{ph}z\|\leq\pi$