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10 Bessel FunctionsBessel and Hankel Functions

§10.22 Integrals

Contents
  1. §10.22(i) Indefinite Integrals
  2. §10.22(ii) Integrals over Finite Intervals
  3. §10.22(iii) Integrals over the Interval (x,)
  4. §10.22(iv) Integrals over the Interval (0,)
  5. §10.22(v) Hankel Transform
  6. §10.22(vi) Compendia

§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)dz =zν+1𝒞ν+1(z),
zν+1𝒞ν(z)dz =zν+1𝒞ν1(z).
10.22.2 zν𝒞ν(z)dz=π122ν1Γ(ν+12)z(𝒞ν(z)𝐇ν1(z)𝒞ν1(z)𝐇ν(z)),
ν12.

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

10.22.3 eizzν𝒞ν(z)dz =eizzν+12ν+1(𝒞ν(z)i𝒞ν+1(z)),
ν12,
eizzν𝒞ν(z)dz =eizzν+112ν(𝒞ν(z)+i𝒞ν1(z)),
ν12.

Products

10.22.4 z𝒞μ(az)𝒟μ(bz)dz=z(a𝒞μ+1(az)𝒟μ(bz)b𝒞μ(az)𝒟μ+1(bz))a2b2,
a2b2,
10.22.5 z𝒞μ(az)𝒟μ(az)dz =14z2(2𝒞μ(az)𝒟μ(az)𝒞μ1(az)𝒟μ+1(az)𝒞μ+1(az)𝒟μ1(az)),
10.22.6 𝒞μ(az)𝒟ν(az)dzz =az(𝒞μ+1(az)𝒟ν(az)𝒞μ(az)𝒟ν+1(az))μ2ν2+𝒞μ(az)𝒟ν(az)μ+ν,
μ2ν2,
10.22.7 zμ+ν+1𝒞μ(az)𝒟ν(az)dz =zμ+ν+22(μ+ν+1)(𝒞μ(az)𝒟ν(az)+𝒞μ+1(az)𝒟ν+1(az)),
μ+ν1,
zμν+1𝒞μ(az)𝒟ν(az)dz =zμν+22(1μν)(𝒞μ(az)𝒟ν(az)+𝒞μ1(az)𝒟ν1(az)),
μ+ν1.

§10.22(ii) Integrals over Finite Intervals

Throughout this subsection x>0.

10.22.8 0xJν(t)dt=2k=0Jν+2k+1(x),
ν>1.
10.22.9 0xJ2n(t)dt=0xJ0(t)dt2k=0n1J2k+1(x),0xJ2n+1(t)dt=1J0(x)2k=1nJ2k(x),
n=0,1,.
10.22.10 0xtμJν(t)dt=xμΓ(12ν+12μ+12)Γ(12ν12μ+12)k=0(ν+2k+1)Γ(12ν12μ+12+k)Γ(12ν+12μ+32+k)Jν+2k+1(x),
(μ+ν+1)>0.
10.22.11 0x1J0(t)tdt =12k=1ψ(k+1)ψ(1)k!(12x)kJk(x),
10.22.12 x0x1J0(t)tdt =2k=0(2k+3)(ψ(k+2)ψ(1))J2k+3(x)=x2J1(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)dθ =12πJν+μ(z)Jνμ(z),
ν>12,
10.22.14 0πJ2ν(2zsinθ)cos(2μθ)dθ =πcos(μπ)Jν+μ(z)Jνμ(z),
ν>12,
10.22.15 0πJ2ν(2zsinθ)sin(2μθ)dθ =πsin(μπ)Jν+μ(z)Jνμ(z),
ν>1.
10.22.16 012πJ0(2zsinθ)cos(2nθ)dθ =12πJn2(z),
n=0,1,2,.
10.22.17 012πY2ν(2zcosθ)cos(2μθ)dθ=12πcot(2νπ)Jν+μ(z)Jνμ(z)12πcsc(2νπ)Jμν(z)Jμν(z),
12<ν<12,
10.22.18 012πY0(2zsinθ)cos(2nθ)dθ=12πJn(z)Yn(z),
n=0,1,2,.
10.22.19 012πJμ(zsinθ)(sinθ)μ+1(cosθ)2ν+1dθ=2νΓ(ν+1)zν1Jμ+ν+1(z),
μ>1, ν>1,
10.22.20 012πJμ(zsinθ)(sinθ)μ(cosθ)2μdθ =π122μ1zμΓ(μ+12)Jμ2(12z),
μ>12,
10.22.21 012πYμ(zsinθ)(sinθ)μ(cosθ)2μdθ =π122μ1zμΓ(μ+12)Jμ(12z)Yμ(12z),
μ>12.
10.22.22 012πJμ(zsin2θ)Jν(zcos2θ)(sinθ)2μ+1(cosθ)2ν+1dθ=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)2α1secθdθ =(μ+ν+α)Γ(μ+α)2α1νΓ(μ+1)zαJμ+ν+α(z),
(μ+α)>0, ν>0.
10.22.24 012πJμ(zsin2θ)Jν(zcos2θ)cotθdθ =12μ1Jμ+ν(z),
μ>0,ν>1.
10.22.25 012πJμ(zsinθ)Iν(zcosθ)(tanθ)μ+1dθ =Γ(12ν12μ)(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+1dθ=zμζνJμ+ν+1(ζ2+z2)(ζ2+z2)12(μ+ν+1),
μ>1,ν>1.

Products

10.22.27 0xtJν12(t)dt =2k=0(ν+2k)Jν+2k2(x),
ν>0,
10.22.28 0xt(Jν12(t)Jν+12(t))dt =2νJν2(x),
ν>0,
10.22.29 0xtJ02(t)dt =12x2(J02(x)+J12(x)).
10.22.30 0xJn(t)Jn+1(t)dt=12(1J02(x))k=1nJk2(x)=k=n+1Jk2(x),
n=0,1,2,.

Convolutions

10.22.31 0xJμ(t)Jν(xt)dt=2k=0(1)kJμ+ν+2k+1(x),
μ>1,ν>1.
10.22.32 0xJν(t)J1ν(xt)dt =J0(x)cosx,
1<ν<2.
10.22.33 0xJν(t)Jν(xt)dt =sinx,
|ν|<1.
10.22.34 0xt1Jμ(t)Jν(xt)dt=Jμ+ν(x)μ,
μ>0,ν>1.
10.22.35 0xJμ(t)Jν(xt)dtt(xt)=(μ+ν)Jμ+ν(x)μνx,
μ>0,ν>0.

Fractional Integral

10.22.36 1Γ(α)0x(xt)α1Jν(t)dt=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)dt=12(Jν(jν,))2δ,m,

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)dt=(a2b2+α2ν2)(Jν(α))22α2δ,m,

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)tdt+γ+ln(12x)=0x1J0(t)tdt=k=1(1)k1(12x)2k2k(k!)2,
10.22.40 xY0(t)tdt=1π(ln(12x)+γ)2+π6+2πk=1(1)k(ψ(k+1)+12kln(12x))(12x)2k2k(k!)2,

where γ is Euler’s constant (§5.2(ii)). Compare (10.22.11) and (10.22.12).

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

10.22.41 0Jν(t)dt =1,
ν>1,
10.22.42 0Yν(t)dt =tan(12νπ),
|ν|<1.
10.22.43 0tμJν(t)dt =2μΓ(12ν+12μ+12)Γ(12ν12μ+12),
(μ+ν)>1, μ<12,
10.22.44 0tμYν(t)dt =2μπΓ(12μ+12ν+12)Γ(12μ12ν+12)sin(12μ12ν)π,
(μ±ν)>1, μ<12.
10.22.45 01J0(t)tμdt=πsec(12μπ)2μΓ2(12μ+12),
1<μ<3.
10.22.46 0tν+1Jν(at)(t2+b2)μ+1dt=aμbνμ2μΓ(μ+1)Kνμ(ab),
a>0, b>0, 1<ν<2μ+32.
10.22.47 0tνYν(at)t2+b2dt=bν1Kν(ab),
a>0,b>0,12<ν<52.

For Kν see §10.25(ii).

10.22.48 0Jμ(xcoshϕ)(coshϕ)1μ(sinhϕ)2ν+1dϕ=2νΓ(ν+1)xν1Jμν1(x),
x>0,ν>1,μ>2ν+12.
10.22.49 0tμ1eatJν(bt)dt=(12b)νaμ+νΓ(μ+ν)𝐅(μ+ν2,μ+ν+12;ν+1;b2a2),
(μ+ν)>0,(a±ib)>0,
10.22.50 0tμ1eatYν(bt)dt=cot(νπ)(12b)νΓ(μ+ν)(a2+b2)12(μ+ν)𝐅(μ+ν2,1μ+ν2;ν+1;b2a2+b2)csc(νπ)(12b)νΓ(μν)(a2+b2)12(μν)𝐅(μν2,1μν2;1ν;b2a2+b2),
μ>|ν|,(a±ib)>0.

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

10.22.51 0Jν(bt)exp(p2t2)tν+1dt =bν(2p2)ν+1exp(b24p2),
ν>1, (p2)>0,
10.22.52 0Jν(bt)exp(p2t2)dt =π2pexp(b28p2)Iν/2(b28p2),
ν>1,(p2)>0,
10.22.53 0Y2ν(bt)exp(p2t2)dt=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)tμ1dt=(12b/p)νΓ(12ν+12μ)2pμexp(b24p2)𝐌(12ν12μ+1,ν+1,b24p2),
(μ+ν)>0, (p2)>0.

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

Orthogonality

10.22.55 0t1Jν+2+1(t)Jν+2m+1(t)dt=δ,m2(2+ν+1),
ν++m>1.

Weber–Schafheitlin Discontinuous Integrals, including Special Cases

10.22.56 0Jμ(at)Jν(bt)tλdt=aμΓ(12ν+12μ12λ+12)2λbμλ+1Γ(12ν12μ+12λ+12)𝐅(12(μ+νλ+1),12(μνλ+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)tλdt =(12a)λ1Γ(12μ+12ν12λ+12)Γ(λ)2Γ(12λ+12ν12μ+12)Γ(12λ+12μ12ν+12)Γ(12λ+12μ+12ν+12),
(μ+ν+1)>λ>0.
10.22.58 0Jν(at)Jν(bt)tλdt =(ab)νΓ(ν12λ+12)2λ(a2+b2)ν12λ+12Γ(12λ+12)𝐅(2ν+1λ4,2ν+3λ4;ν+1;4a2b2(a2+b2)2),
ab, (2ν+1)>λ>1.

When μ>1

10.22.59 0eibtJμ(at)dt={exp(iμarcsin(b/a))(a2b2)12,0b<a,iaμexp(12μπi)(b2a2)12(b+(b2a2)12)μ,0<a<b.
10.22.60 0eibtY0(at)dt={(2i/π)(a2b2)12arcsin(b/a),0b<a,(b2a2)12(1+2iπln(ab+(b2a2)12)),0<a<b.

When μ>0,

10.22.61 0t1eibtJμ(at)dt={(1/μ)exp(iμarcsin(b/a)),0ba,aμexp(12μπi)μ(b+(b2a2)12)μ,0<ab.

When ν>μ>1,

10.22.62 0tμν+1Jμ(at)Jν(bt)dt={0,0<b<a,2μν+1aμ(b2a2)νμ1bνΓ(νμ),0<ab.

When μ>0,

10.22.63 0Jμ(at)Jμ1(bt)dt={bμ1aμ,0<b<a,(2b)1,b=a(>0),0,0<a<b.

When n=0,1,2, and μ>n1,

10.22.64 0Jμ+2n+1(at)Jμ(bt)dt={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))dtt={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 0eatJν(bt)Jν(ct)dt =1π(bc)12Qν12(a2+b2+c22bc),
ν>12.
10.22.67 0texp(p2t2)Jν(at)Jν(bt)dt =12p2exp(a2+b24p2)Iν(ab2p2),
ν>1,(p2)>0.
10.22.68 0texp(p2t2)J0(at)Y0(at)dt =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)tdtt2z2 ={12πiJν(bz)Hν(1)(az),a>b12πiJν(az)Hν(1)(bz),b>a},
ν>1,z>0.
10.22.70 0Yν(at)Jν+1(bt)tdtt2z2 =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+2n3, 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)t1μdt =(bc)μ1(sinϕ)μ12(2π)12aμ𝖯ν1212μ(cosϕ),
μ>12,ν>1,|bc|<a<b+c,cosϕ=(b2+c2a2)/(2bc).
10.22.72 0Jμ(at)Jν(bt)Jν(ct)t1μdt =(bc)μ1sin((μν)π)(sinhχ)μ12(12π3)12aμe(μ12)iπQν1212μ(coshχ),
μ>12,ν>1,a>b+c,coshχ=(a2b2c2)/(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(sa)(sb)(sc),
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)t1νdt ={2ν1Aν12π12(abc)νΓ(ν+12),A>0,0,A0.
If |ν|<12, then
10.22.75 0Yν(at)Jν(bt)Jν(ct)t1+νdt ={(abc)ν(A)ν12π122ν+1Γ(12ν),0<a<|bc|,0,|bc|<a<b+c,(abc)ν(A)ν12π122ν+1Γ(12ν),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)12dx.

Hankel’s inversion theorem is given by

10.22.77 f(y)=0g(x)Jν(xy)(xy)12dx.

Sufficient conditions for the validity of (10.22.77) are that 0|f(x)|dx< when ν12, or that 0|f(x)|dx< and 01xν+12|f(x)|dx< 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), Frenzen and Wong (1985a) and Galapon and Martinez (2014).

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).