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10 Bessel FunctionsBessel and Hankel Functions

§10.23 Sums


§10.23(i) Multiplication Theorem

10.23.1 𝒞ν(λz)=λ±νk=0(1)k(λ2-1)k(12z)kk!𝒞ν±k(z),

If 𝒞=J and the upper signs are taken, then the restriction on λ is unnecessary.

§10.23(ii) Addition Theorems

Neumann’s Addition Theorem

10.23.2 𝒞ν(u±v)=k=-𝒞νk(u)Jk(v),

The restriction |v|<|u| is unnecessary when 𝒞=J and ν is an integer. Special cases are:

10.23.3 J02(z)+2k=1Jk2(z)=1,
10.23.4 k=02n(-1)kJk(z)J2n-k(z)+2k=1Jk(z)J2n+k(z)=0,
10.23.5 k=0nJk(z)Jn-k(z)+2k=1(-1)kJk(z)Jn+k(z)=Jn(2z).

Graf’s and Gegenbauer’s Addition Theorems


10.23.6 w =u2+v2-2uvcosα,
u-vcosα =wcosχ,
vsinα =wsinχ,

the branches being continuous and chosen so that wu and χ0 as v0. If u, v are real and positive and 0απ, then w and χ are real and nonnegative, and the geometrical relationship is shown in Figure 10.23.1.

See accompanying text
Figure 10.23.1: Graf’s and Gegenbauer’s addition theorems. Magnify
10.23.7 𝒞ν(w)cossin(νχ)=k=-𝒞ν+k(u)Jk(v)cossin(kα),
10.23.8 𝒞ν(w)wν=2νΓ(ν)k=0(ν+k)𝒞ν+k(u)uνJν+k(v)vνCk(ν)(cosα),
ν0,-1,, |v±α|<|u|,

where Ck(ν)(cosα) is Gegenbauer’s polynomial (§18.3). The restriction |v±α|<|u| is unnecessary in (10.23.7) when 𝒞=J and ν is an integer, and in (10.23.8) when 𝒞=J.

The degenerate form of (10.23.8) when u= is given by

10.23.9 vcosα=Γ(ν)(12v)νk=0(ν+k)kJν+k(v)Ck(ν)(cosα),

See also §10.12.

Partial Fractions

For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005).

§10.23(iii) Series Expansions of Arbitrary Functions

Neumann’s Expansion

10.23.10 f(z)=a0J0(z)+2k=1akJk(z),

where c is the distance of the nearest singularity of the analytic function f(z) from z=0,

10.23.11 ak=12π|z|=cf(t)Ok(t)t,

and Ok(t) is Neumann’s polynomial, defined by the generating function:

10.23.12 1t-z=J0(z)O0(t)+2k=1Jk(z)Ok(t),

On(t) is a polynomial of degree n+1 in 1/t:O0(t)=1/t and

10.23.13 On(t)=14k=0n/2(n-k-1)!nk!(2t)n-2k+1,

For the more general form of expansion

10.23.14 zνf(z)=a0Jν(z)+2k=1akJν+k(z)

see Watson (1944, §16.13), and for further generalizations see Watson (1944, Chapter 16) and Erdélyi et al. (1953b, §7.10.1).


10.23.15 (12z)ν=k=0(ν+2k)Γ(ν+k)k!Jν+2k(z),
10.23.16 Y0(z)=2π(ln(12z)+γ)J0(z)-4πk=1(-1)kJ2k(z)k,
10.23.17 Yn(z)=-n!(12z)-nπk=0n-1(12z)kJk(z)k!(n-k)+2π(ln(12z)-ψ(n+1))Jn(z)-2πk=1(-1)k(n+2k)Jn+2k(z)k(n+k),

where γ is Euler’s constant and ψ(n+1)=Γ(n+1)/Γ(n+1)5.2).

Other examples are provided by (10.12.1)–(10.12.6), (10.23.2), and (10.23.7).

Fourier–Bessel Expansion

Assume f(t) satisfies

10.23.18 01t12|f(t)|t<,

and define

10.23.19 am=2(Jν+1(jν,m))201tf(t)Jν(jν,mt)t,

where jν,m is as in §10.21(i). If 0<x<1, then

10.23.20 12f(x-)+12f(x+)=m=1amJν(jν,mx),

provided that f(t) is of bounded variation (§1.4(v)) on an interval [a,b] with 0<a<x<b<1. This result is proved in Watson (1944, Chapter 18) and further information is provided in this reference, including the behavior of the series near x=0 and x=1.

As an example,

10.23.21 xν=m=12Jν(jν,mx)jν,mJν+1(jν,m),

(Note that when x=1 the left-hand side is 1 and the right-hand side is 0.)

Other Series Expansions

For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). See also Schäfke (1960, 1961b).

§10.23(iv) Compendia

For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).