About the Project
10 Bessel FunctionsModified Bessel Functions

§10.43 Integrals


§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),
z-ν+1𝒵ν(z)dz =z-ν+1𝒵ν-1(z).
10.43.2 zν𝒵ν(z)dz=π122ν-1Γ(ν+12)z(𝒵ν(z)Lν-1(z)-𝒵ν-1(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±zz-ν𝒵ν(z)dz =e±zz-ν+11-2ν(𝒵ν(z)𝒵ν-1(z)),

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

10.43.4 0xI0(t)-1tdt=12k=1(-1)k-1ψ(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+π224-k=1(ψ(k+1)+12k-ln(12x))(12x)2k2k(k!)2,

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

10.43.6 0xe-tIn(t)dt=xe-x(I0(x)+I1(x))+n(e-xI0(x)-1)+2e-xk=1n-1(n-k)Ik(x),
10.43.7 0xe±ttνIν(t)dt=e±xxν+12ν+1(Iν(x)Iν+1(x)),
10.43.8 0xe±tt-νIν(t)dt=-e±xx-ν+12ν-1(Iν(x)Iν-1(x))2-ν+1(2ν-1)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 xett-νKν(t)dt=exx-ν+12ν-1(Kν(x)+Kν-1(x)),

§10.43(iii) Fractional Integrals

The Bickley function Kiα(x) is defined by

10.43.11 Kiα(x)=1Γ(α)x(t-x)α-1K0(t)dt,

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

10.43.12 Kiα(x)=0e-xcosht(cosht)αdt,


10.43.13 Kiα(x)=xKiα-1(t)dt,
10.43.14 Ki0(x)=K0(x),
10.43.15 Ki-n(x)=(-1)ndndxnK0(x),
10.43.16 Kiα(0)=πΓ(12α)2Γ(12α+12),
10.43.17 αKiα+1(x)+xKiα(x)+(1-α)Kiα-1(x)-xKiα-2(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 0tμ-1Kν(t)dt=2μ-2Γ(12μ-12ν)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 0tμ-1e-atKν(t)dt={(12π)12Γ(μ-ν)Γ(μ+ν)(1-a2)-12μ+14Pν-12-μ+12(a),-1<a<1,(12π)12Γ(μ-ν)Γ(μ+ν)(a2-1)-12μ+14Pν-12-μ+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(12ν-12λ+12μ+12)Γ(12ν-12λ-12μ+12)2λ+1aν-λ+1F(ν-λ+μ+12,ν-λ-μ+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<, x-1g(x) is continuously differentiable and each of xg(x) and xd(x-1g(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).