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10 Bessel FunctionsBessel and Hankel Functions

§10.22 Integrals

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§10.22(i) Indefinite Integrals

In this subsection 𝒞ν(z) and 𝒟μ(z) denote cylinder functions(§10.2(ii)) of orders ν and μ, respectively, not necessarily distinct.

10.22.1 zν+1𝒞ν(z)z =zν+1𝒞ν+1(z),
z-ν+1𝒞ν(z)z =-z-ν+1𝒞ν-1(z).
10.22.2 zν𝒞ν(z)z=π122ν-1Γ(ν+12)z(𝒞ν(z)Hν-1(z)-𝒞ν-1(z)Hν(z)),
ν-12.

For the Struve function Hν(z) see §11.2(i).

10.22.3 zzν𝒞ν(z)z =zzν+12ν+1(𝒞ν(z)-𝒞ν+1(z)),
ν-12,
zz-ν𝒞ν(z)z =zz-ν+11-2ν(𝒞ν(z)+𝒞ν-1(z)),
ν12.

Products

10.22.4 z𝒞μ(az)𝒟μ(bz)z=z(a𝒞μ+1(az)𝒟μ(bz)-b𝒞μ(az)𝒟μ+1(bz))a2-b2,
a2b2,
10.22.5 z𝒞μ(az)𝒟μ(az)z =14z2(2𝒞μ(az)𝒟μ(az)-𝒞μ-1(az)𝒟μ+1(az)-𝒞μ+1(az)𝒟μ-1(az)),
10.22.6 𝒞μ(az)𝒟ν(az)zz =-az(𝒞μ+1(az)𝒟ν(az)-𝒞μ(az)𝒟ν+1(az))μ2-ν2+𝒞μ(az)𝒟ν(az)μ+ν,
μ2ν2,
10.22.7 zμ+ν+1𝒞μ(az)𝒟ν(az)z =zμ+ν+22(μ+ν+1)(𝒞μ(az)𝒟ν(az)+𝒞μ+1(az)𝒟ν+1(az)),
μ+ν-1,
z-μ-ν+1𝒞μ(az)𝒟ν(az)z =z-μ-ν+22(1-μ-ν)(𝒞μ(az)𝒟ν(az)+𝒞μ-1(az)𝒟ν-1(az)),
μ+ν1.

§10.22(ii) Integrals over Finite Intervals

Throughout this subsection x>0.

10.22.8 0xJν(t)t=2k=0Jν+2k+1(x),
ν>-1.
10.22.9 0xJ2n(t)t=0xJ0(t)t-2k=0n-1J2k+1(x), 0xJ2n+1(t)t=1-J0(x)-2k=1nJ2k(x),
n=0,1,.
10.22.10 0xtμJν(t)t=xμΓ(12ν+12μ+12)Γ(12ν-12μ+12)k=0(ν+2k+1)Γ(12ν-12μ+12+k)Γ(12ν+12μ+32+k)Jν+2k+1(x),
(μ+ν+1)>0.
10.22.11 0x1-J0(t)tt =12k=1ψ(k+1)-ψ(1)k!(12x)kJk(x),
10.22.12 x0x1-J0(t)tt =2k=0(2k+3)(ψ(k+2)-ψ(1))J2k+3(x)
=x-2J1(x)+2k=0(2k+5)(ψ(k+3)-ψ(1)-1)J2k+5(x),

where ψ(x)=Γ(x)/Γ(x)5.2(i)). See also (10.22.39).

Trigonometric Arguments

10.22.13 012πJ2ν(2zcosθ)cos(2μθ)θ =12πJν+μ(z)Jν-μ(z),
ν>-12,
10.22.14 0πJ2ν(2zsinθ)cos(2μθ)θ =πcos(μπ)Jν+μ(z)Jν-μ(z),
ν>-12,
10.22.15 0πJ2ν(2zsinθ)sin(2μθ)θ =πsin(μπ)Jν+μ(z)Jν-μ(z),
ν>-1.
10.22.16 012πJ0(2zsinθ)cos(2nθ)θ =12πJn2(z),
n=0,1,2,.
10.22.17 012πY2ν(2zcosθ)cos(2μθ)θ=12πcot(2νπ)Jν+μ(z)Jν-μ(z)-12πcsc(2νπ)Jμ-ν(z)J-μ-ν(z),
-12<ν<12,
10.22.18 012πY0(2zsinθ)cos(2nθ)θ=12πJn(z)Yn(z),
n=0,1,2,.
10.22.19 012πJμ(zsinθ)(sinθ)μ+1(cosθ)2ν+1θ=2νΓ(ν+1)z-ν-1Jμ+ν+1(z),
μ>-1, ν>-1,
10.22.20 012πJμ(zsinθ)(sinθ)μ(cosθ)2μθ =π122μ-1z-μΓ(μ+12)Jμ2(12z),
μ>-12,
10.22.21 012πYμ(zsinθ)(sinθ)μ(cosθ)2μθ =π122μ-1z-μΓ(μ+12)Jμ(12z)Yμ(12z),
μ>-12.
10.22.22 012πJμ(zsin2θ)Jν(zcos2θ)(sinθ)2μ+1(cosθ)2ν+1θ=Γ(μ+12)Γ(ν+12)Jμ+ν+12(z)(8πz)12Γ(μ+ν+1),
μ>-12,ν>-12.
10.22.23 012πJμ(zsin2θ)Jν(zcos2θ)(sinθ)2α-1secθθ =(μ+ν+α)Γ(μ+α)2α-1νΓ(μ+1)zαJμ+ν+α(z),
(μ+α)>0, ν>0.
10.22.24 012πJμ(zsin2θ)Jν(zcos2θ)cotθθ =12μ-1Jμ+ν(z),
μ>0,ν>-1.
10.22.25 012πJμ(zsinθ)Iν(zcosθ)(tanθ)μ+1θ =Γ(12ν-12μ)(12z)μ2Γ(12ν+12μ+1)Jν(z),
ν>μ>-1.

For Iν see §10.25(ii).

10.22.26 012πJμ(zsinθ)Jν(ζcosθ)(sinθ)μ+1(cosθ)ν+1θ=zμζνJμ+ν+1(ζ2+z2)(ζ2+z2)12(μ+ν+1),
μ>-1,ν>-1.

Products

10.22.27 0xtJν-12(t)t =2k=0(ν+2k)Jν+2k2(x),
ν>0,
10.22.28 0xt(Jν-12(t)-Jν+12(t))t =2νJν2(x),
ν>0,
10.22.29 0xtJ02(t)t =12x2(J02(x)+J12(x)).
10.22.30 0xJn(t)Jn+1(t)t=12(1-J02(x))-k=1nJk2(x)=k=n+1Jk2(x),
n=0,1,2,.

Convolutions

10.22.31 0xJμ(t)Jν(x-t)t=2k=0(-1)kJμ+ν+2k+1(x),
μ>-1,ν>-1.
10.22.32 0xJν(t)J1-ν(x-t)t =J0(x)-cosx,
-1<ν<2.
10.22.33 0xJν(t)J-ν(x-t)t =sinx,
|ν|<1.
10.22.34 0xt-1Jμ(t)Jν(x-t)t=Jμ+ν(x)μ,
μ>0,ν>-1.
10.22.35 0xJμ(t)Jν(x-t)tt(x-t)=(μ+ν)Jμ+ν(x)μνx,
μ>0,ν>0.

Fractional Integral

10.22.36 1Γ(α)0x(x-t)α-1Jν(t)t=2αk=0(α)kk!Jν+α+2k(x),
α>0,ν0.

When α=m=1,2,3, the left-hand side of (10.22.36) is the mth repeated integral of Jν(x) (§§1.4(v) and 1.15(vi)).

Orthogonality

If ν>-1, then

10.22.37 01tJν(jν,t)Jν(jν,mt)t=12δ,m(Jν(jν,))2,

where jν, and jν,m are zeros of Jν(x)10.21(i)), and δ,m is Kronecker’s symbol.

Also, if a,b,ν are real constants with b0 and ν>-1, then

10.22.38 01tJν(αt)Jν(αmt)t=δ,m(a2b2+α2-ν2)(Jν(α))22α2,

where α and αm are positive zeros of aJν(x)+bxJν(x). (Compare (10.22.55)).

§10.22(iii) Integrals over the Interval (x,)

When x>0

10.22.39 xJ0(t)tt+γ+ln(12x)=0x1-J0(t)tt=k=1(-1)k-1(12x)2k2k(k!)2,
10.22.40 xY0(t)tt=-1π(ln(12x)+γ)2+π6+2πk=1(-1)k(ψ(k+1)+12k-ln(12x))(12x)2k2k(k!)2,

where γ is Euler’s constant (§5.2(ii)).

§10.22(iv) Integrals over the Interval (0,)

10.22.41 0Jν(t)t =1,
ν>-1,
10.22.42 0Yν(t)t =-tan(12νπ),
|ν|<1.
10.22.43 0tμJν(t)t =2μΓ(12ν+12μ+12)Γ(12ν-12μ+12),
(μ+ν)>-1, μ<12,
10.22.44 0tμYν(t)t =2μπΓ(12μ+12ν+12)Γ(12μ-12ν+12)sin(12μ-12ν)π,
(μ±ν)>-1, μ<12.
10.22.45 01-J0(t)tμt=-πsec(12μπ)2μΓ2(12μ+12),
1<μ<3.
10.22.46 0tν+1Jν(at)(t2+b2)μ+1t=aμbν-μ2μΓ(μ+1)Kν-μ(ab),
a>0, b>0, -1<ν<2μ+32.
10.22.47 0tνYν(at)t2+b2t=-bν-1Kν(ab),
a>0,b>0,-12<ν<52.

For Kν see §10.25(ii).

10.22.48 0Jμ(xcoshϕ)(coshϕ)1-μ(sinhϕ)2ν+1ϕ=2νΓ(ν+1)x-ν-1Jμ-ν-1(x),
x>0,ν>-1,μ>2ν+12.
10.22.49 0tμ-1-atJν(bt)t=(12b)νaμ+νΓ(μ+ν)F(μ+ν2,μ+ν+12;ν+1;-b2a2),
(μ+ν)>0,(a±b)>0,
10.22.50 0tμ-1-atYν(bt)t=cot(νπ)(12b)νΓ(μ+ν)(a2+b2)12(μ+ν)F(μ+ν2,1-μ+ν2;ν+1;b2a2+b2)-csc(νπ)(12b)-νΓ(μ-ν)(a2+b2)12(μ-ν)F(μ-ν2,1-μ-ν2;1-ν;b2a2+b2),
μ>|ν|,(a±b)>0.

For the hypergeometric function F see §15.2(i).

10.22.51 0Jν(bt)exp(-p2t2)tν+1t =bν(2p2)ν+1exp(-b24p2),
ν>-1, (p2)>0,
10.22.52 0Jν(bt)exp(-p2t2)t =π2pexp(-b28p2)Iν/2(b28p2),
ν>-1,(p2)>0,
10.22.53 0Y2ν(bt)exp(-p2t2)t=-π2pexp(-b28p2)(Iν(b28p2)tan(νπ)+1πKν(b28p2)sec(νπ)),
|ν|<12, (p2)>0.

For I and K see §10.25(ii).

10.22.54 0Jν(bt)exp(-p2t2)tμ-1t=(12b/p)νΓ(12ν+12μ)2pμexp(-b24p2)M(12ν-12μ+1,ν+1,b24p2),
(μ+ν)>0, (p2)>0.

For the confluent hypergeometric function M see §13.2(i).

Orthogonality

10.22.55 0t-1Jν+2+1(t)Jν+2m+1(t)t=δ,m2(2+ν+1),
ν++m>-1.

Weber–Schafheitlin Discontinuous Integrals, including Special Cases

10.22.56 0Jμ(at)Jν(bt)tλt=aμΓ(12ν+12μ-12λ+12)2λbμ-λ+1Γ(12ν-12μ+12λ+12)F(12(μ+ν-λ+1),12(μ-ν-λ+1);μ+1;a2b2),
0<a<b, (μ+ν+1)>λ>-1.

If 0<b<a, then interchange a and b, and also μ and ν. If b=a, then

10.22.57 0Jμ(at)Jν(at)tλt =(12a)λ-1Γ(12μ+12ν-12λ+12)Γ(λ)2Γ(12λ+12ν-12μ+12)Γ(12λ+12μ-12ν+12)Γ(12λ+12μ+12ν+12),
(μ+ν+1)>λ>0.
10.22.58 0Jν(at)Jν(bt)tλt =(ab)νΓ(ν-12λ+12)2λ(a2+b2)ν-12λ+12Γ(12λ+12)F(2ν+1-λ4,2ν+3-λ4;ν+1;4a2b2(a2+b2)2),
ab, (2ν+1)>λ>-1.

When μ>-1

10.22.59 0btJμ(at)t={exp(μarcsin(b/a))(a2-b2)12,0b<a,aμexp(12μπ)(b2-a2)12(b+(b2-a2)12)μ,0<a<b.
10.22.60 0btY0(at)t={(2/π)(a2-b2)-12arcsin(b/a),0b<a,(b2-a2)-12(-1+2πln(ab+(b2-a2)12)),0<a<b.

When μ>0,

10.22.61 0t-1btJμ(at)t={(1/μ)exp(μarcsin(b/a)),0ba,aμexp(12μπ)μ(b+(b2-a2)12)μ,0<ab.

When ν>μ>-1,

10.22.62 0tμ-ν+1Jμ(at)Jν(bt)t={0,0<b<a,2μ-ν+1aμ(b2-a2)ν-μ-1bνΓ(ν-μ),0<ab.

When μ>0,

10.22.63 0Jμ(at)Jμ-1(bt)t={bμ-1a-μ,0<b<a,(2b)-1,b=a(>0),0,0<a<b.

When n=0,1,2, and μ>-n-1,

10.22.64 0Jμ+2n+1(at)Jμ(bt)t={bμΓ(μ+n+1)aμ+1n!F(-n,μ+n+1;μ+1;b2a2),0<b<a,(-1)n/(2a),b=a(>0),0,0<a<b.
10.22.65 0J0(at)(J0(bt)-J0(ct))tt={0,0b<a,0<ca,ln(c/a),0b<ac.

Other Double Products

In (10.22.66)–(10.22.70) a,b,c are positive constants.

10.22.66 0-atJν(bt)Jν(ct)t =1π(bc)12Qν-12(a2+b2+c22bc),
ν>-12.
10.22.67 0texp(-p2t2)Jν(at)Jν(bt)t =12p2exp(-a2+b24p2)Iν(ab2p2),
ν>-1,(p2)>0.
10.22.68 0texp(-p2t2)J0(at)Y0(at)t =-12πp2exp(-a22p2)K0(a22p2),
(p2)>0.

For the associated Legendre function Q see §14.3(ii) with μ=0. For I and K see §10.25(ii).

10.22.69 0Jν(at)Jν(bt)ttt2-z2 ={12πJν(bz)Hν(1)(az),a>b12πJν(az)Hν(1)(bz),b>a},
ν>-1,z>0.
10.22.70 0Yν(at)Jν+1(bt)ttt2-z2 =12πJν+1(bz)Hν(1)(az),
ab>0, ν>-32,z>0.

Equation (10.22.70) also remains valid if the order ν+1 of the J functions on both sides is replaced by ν+2n-3, n=1,2,, and the constraint ν>-32 is replaced by ν>-n+12.

See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions.

Triple Products

In (10.22.71) and (10.22.72) a,b,c are positive constants.

10.22.71 0Jμ(at)Jν(bt)Jν(ct)t1-μt =(bc)μ-1(sinϕ)μ-12(2π)12aμPν-1212-μ(cosϕ),
μ>-12,ν>-1,|b-c|<a<b+c,cosϕ=(b2+c2-a2)/(2bc).
10.22.72 0Jμ(at)Jν(bt)Jν(ct)t1-μt =(bc)μ-1cos(νπ)(sinhχ)μ-12(12π3)12aμQν-1212-μ(coshχ),
μ>-12,ν>-1,a>b+c,coshχ=(a2-b2-c2)/(2bc).

For the Ferrers function P and the associated Legendre function Q, see §§14.3(i) and 14.3(ii), respectively.

In (10.22.74) and (10.22.75), a,b,c are positive constants and

10.22.73 A =s(s-a)(s-b)(s-c),
s =12(a+b+c).

(Thus if a,b,c are the sides of a triangle, then A12 is the area of the triangle.)

If ν>-12, then

10.22.74 0Jν(at)Jν(bt)Jν(ct)t1-νt ={2ν-1Aν-12π12(abc)νΓ(ν+12),A>0,0,A0.
If |ν|<12, then
10.22.75 0Yν(at)Jν(bt)Jν(ct)t1+νt ={-(abc)ν(-A)-ν-12π122ν+1Γ(12-ν),0<a<|b-c|,0,|b-c|<a<b+c,(abc)ν(-A)-ν-12π122ν+1Γ(12-ν),a>b+c.

Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b).

§10.22(v) Hankel Transform

The Hankel transform (or Bessel transform) of a function f(x) is defined as

10.22.76 g(y)=0f(x)Jν(xy)(xy)12x.

Hankel’s inversion theorem is given by

10.22.77 f(y)=0g(x)Jν(xy)(xy)12x.

Sufficient conditions for the validity of (10.22.77) are that 0|f(x)|x< when ν-12, or that 0|f(x)|x< and 01xν+12|f(x)|x< when -1<ν<-12; see Titchmarsh (1986a, Theorem 135, Chapter 8) and Akhiezer (1988, p. 62).

For asymptotic expansions of Hankel transforms see Wong (1976, 1977) and Frenzen and Wong (1985).

For collections of Hankel transforms see Erdélyi et al. (1954b, Chapter 8) and Oberhettinger (1972).

§10.22(vi) Compendia

For collections of integrals of the functions Jν(z), Yν(z), Hν(1)(z), and Hν(2)(z), including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5 and 6.5–6.7), Gröbner and Hofreiter (1950, pp. 196–204), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1974, §§1.10 and 2.7), Oberhettinger (1990, §§1.13–1.16 and 2.13–2.16), Oberhettinger and Badii (1973, §§1.14 and 2.12), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.8–1.10, 2.12–2.14, 3.2.4–3.2.7, 3.3.2, and 3.4.1), Prudnikov et al. (1992a, §§3.12–3.14), Prudnikov et al. (1992b, §§3.12–3.14), Watson (1944, Chapters 5, 12, 13, and 14), and Wheelon (1968).