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

§10.2 Definitions

Keywords:: Bessel functions
Referenced by:: §10.74(i), §2.8(i), §32.10(iii)
Permalink:: http://dlmf.nist.gov/10.2

§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

Keywords:: Bessel functions, Bessel’s equation, Hankel functions, cylinder functions, differential equations, singularities
Permalink:: http://dlmf.nist.gov/10.2.i

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

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

Notes:: See Olver (1997b, pp. 57, 237–238, 242–243) and Watson (1944, pp. 38–45, 57–64, 196–198).
Keywords:: Bessel functions, Bessel’s equation
Referenced by:: ¶ ‣ §1.17(ii), §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

¶ Bessel Function of the First Kind

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

Defines:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : : factorial, : 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, §8.6(i)
Permalink:: http://dlmf.nist.gov/10.2.E2
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This solution of (10.2.1) is an analytic function of $z\in\Complex$ , except for a branch point at when $\nu$ is not an integer. The principal branch of $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ 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 $\nu=n$ $(\in\Integer)$ , $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ is entire in .

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

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

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

Defines:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind
Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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
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When $\nu$ is an integer the right-hand side is replaced by its limiting value:

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : 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
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Whether or not $\nu$ is an integer $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ has a branch point at . The principal branch corresponds to the principal branches of $\mathop{J_{{\pm\nu}}\/}\nolimits\!\left(z\right)$ in (10.2.3) and (10.2.4), with a cut in the -plane along the interval $(-\infty,0]$ .

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

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

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

¶ Bessel Functions of the Third Kind (Hankel Functions)

Keywords:: Hankel functions

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

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

Defines:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function)
Symbols:: : base of exponential function, $\sim$ : asymptotic equality, : 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
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as $z\to\infty$ in $-\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq 2\pi-\delta$ , and

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

Defines:: $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function)
Symbols:: : base of exponential function, $\sim$ : asymptotic equality, : 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
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as $z\to\infty$ in $-2\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi-\delta$ , where $\delta$ is an arbitrary small positive constant. Each solution has a branch point at for all $\nu\in\Complex$ . 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 $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ and $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ are two-valued and discontinuous on the cut $\mathop{\mathrm{ph}\/}\nolimits z=\pm\pi$ .

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

¶ Branch Conventions

Keywords:: Bessel functions, Hankel functions

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

¶ Cylinder Functions

Defines:: $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ : cylinder function
Keywords:: Bessel functions, Bessel’s equation, cylinder functions

The notation $\mathop{\mathscr{C}_{{\nu}}\/}\nolimits\!\left(z\right)$ denotes $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ , $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ , $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ , 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

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.
Keywords:: Bessel functions, Bessel’s equation
Permalink:: http://dlmf.nist.gov/10.2.iii

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 $\realpart{\nu}\geq 0$ . When $\realpart{\nu}<0$ , $\nu$ is replaced by $-\nu$ throughout.

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

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