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10 Bessel FunctionsModified Bessel Functions

§10.43 Integrals

  1. §10.43(i) Indefinite Integrals
  2. §10.43(ii) Integrals over the Intervals (0,x) and (x,)
  3. §10.43(iii) Fractional Integrals
  4. §10.43(iv) Integrals over the Interval (0,)
  5. §10.43(v) Kontorovich–Lebedev Transform
  6. §10.43(vi) Compendia

§10.43(i) Indefinite Integrals

Let 𝒵ν(z) be defined as in §10.25(ii). Then

10.43.1 zν+1𝒵ν(z)dz =zν+1𝒵ν+1(z),
zν+1𝒵ν(z)dz =zν+1𝒵ν1(z).
10.43.2 zν𝒵ν(z)dz=π122ν1Γ(ν+12)z(𝒵ν(z)𝐋ν1(z)𝒵ν1(z)𝐋ν(z)),

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

10.43.3 e±zzν𝒵ν(z)dz =e±zzν+12ν+1(𝒵ν(z)𝒵ν+1(z)),
e±zzν𝒵ν(z)dz =e±zzν+112ν(𝒵ν(z)𝒵ν1(z)),

§10.43(ii) Integrals over the Intervals (0,x) and (x,)

10.43.4 0xI0(t)1tdt=12k=1(1)k1ψ(k+1)ψ(1)k!(12x)kIk(x)=2xk=0(1)k(2k+3)(ψ(k+2)ψ(1))I2k+3(x).
10.43.5 xK0(t)tdt=12(ln(12x)+γ)2+π224k=1(ψ(k+1)+12kln(12x))(12x)2k2k(k!)2,

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

10.43.6 0xetIn(t)dt=xex(I0(x)+I1(x))+n(exI0(x)1)+2exk=1n1(nk)Ik(x),
10.43.7 0xe±ttνIν(t)dt=e±xxν+12ν+1(Iν(x)Iν+1(x)),
10.43.8 0xe±ttνIν(t)dt=e±xxν+12ν1(Iν(x)Iν1(x))2ν+1(2ν1)Γ(ν),
10.43.9 0xe±ttνKν(t)dt=e±xxν+12ν+1(Kν(x)±Kν+1(x))2νΓ(ν+1)2ν+1,
10.43.10 xettνKν(t)dt=exxν+12ν1(Kν(x)+Kν1(x)),

§10.43(iii) Fractional Integrals

The Bickley function Kiα(x) is defined by

10.43.11 Kiα(x)=1Γ(α)x(tx)α1K0(t)dt,

when α>0 and x>0, and by analytic continuation elsewhere. Equivalently,

10.43.12 Kiα(x)=0excosht(cosht)αdt,


10.43.13 Kiα(x)=xKiα1(t)dt,
10.43.14 Ki0(x)=K0(x),
10.43.15 Kin(x)=(1)ndndxnK0(x),
10.43.16 Kiα(0)=πΓ(12α)2Γ(12α+12),
10.43.17 αKiα+1(x)+xKiα(x)+(1α)Kiα1(x)xKiα2(x)=0.

For further properties of the Bickley function, including asymptotic expansions and generalizations, see Amos (1983c, 1989) and Luke (1962, Chapter 8).

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

10.43.18 0Kν(t)dt=12πsec(12πν),
10.43.19 0tμ1Kν(t)dt=2μ2Γ(12μ12ν)Γ(12μ+12ν),
10.43.20 0cos(at)K0(t)dt =π2(1+a2)12,
10.43.21 0sin(at)K0(t)dt =arcsinha(1+a2)12,

When μ>|ν|,

10.43.22 0tμ1eatKν(t)dt={(12π)12Γ(μν)Γ(μ+ν)(1a2)12μ+14𝖯ν12μ+12(a),1<a<1,(12π)12Γ(μν)Γ(μ+ν)(a21)12μ+14Pν12μ+12(a),a0,a1.

For the second equation there is a cut in the a-plane along the interval [0,1], and all quantities assume their principal values (§4.2(i)). For the Ferrers function 𝖯 and the associated Legendre function P, see §§14.3(i) and 14.21(i).

10.43.23 0tν+1Iν(bt)exp(p2t2)dt =bν(2p2)ν+1exp(b24p2),
10.43.24 0Iν(bt)exp(p2t2)dt =π2pexp(b28p2)I12ν(b28p2),
ν>1, (p2)>0,
10.43.25 0Kν(bt)exp(p2t2)dt =π4psec(12πν)exp(b28p2)K12ν(b28p2),
|ν|<1, (p2)>0.
10.43.26 0Kμ(at)Jν(bt)tλdt =bνΓ(12ν12λ+12μ+12)Γ(12ν12λ12μ+12)2λ+1aνλ+1𝐅(νλ+μ+12,νλμ+12;ν+1;b2a2),

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

10.43.27 0tμ+ν+1Kμ(at)Jν(bt)dt =(2a)μ(2b)νΓ(μ+ν+1)(a2+b2)μ+ν+1,
10.43.28 0texp(p2t2)Iν(at)Iν(bt)dt =12p2exp(a2+b24p2)Iν(ab2p2),
10.43.29 0texp(p2t2)I0(at)K0(at)dt =14p2exp(a22p2)K0(a22p2),

For infinite integrals of triple products of modified and unmodified Bessel functions, see Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b).

§10.43(v) Kontorovich–Lebedev Transform

The Kontorovich–Lebedev transform of a function g(x) is defined as

10.43.30 f(y)=2yπ2sinh(πy)0g(x)xKiy(x)dx.


10.43.31 g(x)=0f(y)Kiy(x)dy,

provided that either of the following sets of conditions is satisfied:

  • (a)

    On the interval 0<x<, x1g(x) is continuously differentiable and each of xg(x) and xd(x1g(x))/dx is absolutely integrable.

  • (b)

    g(x) is piecewise continuous and of bounded variation on every compact interval in (0,), and each of the following integrals

10.43.32 012g(x)xln(1x)dx,
  • converges.

For asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996).

For collections of the Kontorovich–Lebedev transform, see Erdélyi et al. (1954b, Chapter 12), Prudnikov et al. (1986b, pp. 404–412), and Oberhettinger (1972, Chapter 5).

§10.43(vi) Compendia

For collections of integrals of the functions Iν(z) and Kν(z), including integrals with respect to the order, see 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, 6.5–6.7), Gröbner and Hofreiter (1950, pp. 197–203), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1972), Oberhettinger (1974, §§1.11 and 2.7), Oberhettinger (1990, §§1.17–1.20 and 2.17–2.20), Oberhettinger and Badii (1973, §§1.15 and 2.13), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.11–1.12, 2.15–2.16, 3.2.8–3.2.10, and 3.4.1), Prudnikov et al. (1992a, §§3.15, 3.16), Prudnikov et al. (1992b, §§3.15, 3.16), Watson (1944, Chapter 13), and Wheelon (1968).