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10 Bessel FunctionsBessel and Hankel Functions

§10.2 Definitions

Contents

§10.2(i) Bessel’s Equation

10.2.1 z2d2wdz2+zdwdz+(z2-ν2)w=0.

This differential equation has a regular singularity at z=0 with indices ±ν, and an irregular singularity at z= of rank 1; compare §§2.7(i) and 2.7(ii).

§10.2(ii) Standard Solutions

Bessel Function of the First Kind

10.2.2 Jν(z)=(12z)νk=0(-1)k(14z2)kk!Γ(ν+k+1).

This solution of (10.2.1) is an analytic function of z, except for a branch point at z=0 when ν is not an integer. The principal branch of Jν(z) corresponds to the principal value of (12z)ν4.2(iv)) and is analytic in the z-plane cut along the interval (-,0].

When ν=n (), Jν(z) is entire in z.

For fixed z (0) each branch of Jν(z) is entire in ν.

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

10.2.3 Yν(z)=Jν(z)cos(νπ)-J-ν(z)sin(νπ).

When ν is an integer the right-hand side is replaced by its limiting value:

10.2.4 Yn(z)=1πJν(z)ν|ν=n+(-1)nπJν(z)ν|ν=-n,
n=0,±1,±2,.

Whether or not ν is an integer Yν(z) has a branch point at z=0. The principal branch corresponds to the principal branches of J±ν(z) in (10.2.3) and (10.2.4), with a cut in the z-plane along the interval (-,0].

Except in the case of J±n(z), the principal branches of Jν(z) and Yν(z) are two-valued and discontinuous on the cut phz=±π; compare §4.2(i).

Both Jν(z) and Yν(z) are real when ν is real and phz=0.

For fixed z (0) each branch of Yν(z) is entire in ν.

Bessel Functions of the Third Kind (Hankel Functions)

These solutions of (10.2.1) are denoted by Hν(1)(z) and Hν(2)(z), and their defining properties are given by

10.2.5 Hν(1)(z)2/(πz)ei(z-12νπ-14π)

as z in -π+δphz2π-δ, and

10.2.6 Hν(2)(z)2/(πz)e-i(z-12νπ-14π)

as z in -2π+δphzπ-δ, where δ is an arbitrary small positive constant. Each solution has a branch point at z=0 for all ν. 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 z-plane along the interval (-,0].

The principal branches of Hν(1)(z) and Hν(2)(z) are two-valued and discontinuous on the cut phz=±π.

For fixed z (0) each branch of Hν(1)(z) and Hν(2)(z) is entire in ν.

Branch Conventions

Except where indicated otherwise, it is assumed throughout the DLMF that the symbols Jν(z), Yν(z), Hν(1)(z), and Hν(2)(z) denote the principal values of these functions.

Cylinder Functions

The notation 𝒞ν(z) denotes Jν(z), Yν(z), Hν(1)(z), Hν(2)(z), or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν.

§10.2(iii) Numerically Satisfactory Pairs of Solutions

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 ν0. When ν<0, ν is replaced by -ν throughout.

Table 10.2.1: Numerically satisfactory pairs of solutions of Bessel’s equation.
Pair Interval or Region
Jν(x),Yν(x) 0<x<
Jν(z),Yν(z) neighborhood of 0 in |phz|π
Jν(z),Hν(1)(z) 0phzπ
Jν(z),Hν(2)(z) -πphz0
Hν(1)(z),Hν(2)(z) neighborhood of in |phz|π