§10.43 Integrals ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions (original) (raw)

Contents

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

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

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

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

§10.43(iii) Fractional Integrals

The Bickley function Kiα⁡(x) is defined by

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

Properties

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

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

When ℜ⁡μ>|ℜ⁡ν|,

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 functionP, see §§14.3(i) and 14.21(i).

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

For infinite integrals of triple products of modified and unmodified Bessel functions, seeGervois 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

Then

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

• (a)
  On the interval 0<x<∞, x−1⁢g⁡(x) is continuously differentiable and each of x⁢g⁡(x) and x⁢d(x−1⁢g⁡(x))/dxis absolutely integrable.
• (b)
  g⁡(x) is piecewise continuous and of bounded variation on every compact interval in (0,∞), and each of the following integrals

• converges.

For asymptotic expansions of the direct transform (10.43.30) seeWong (1981), and for asymptotic expansions of the inverse transform (10.43.31) seeNaylor (1990, 1996).

For collections of the Kontorovich–Lebedev transform, seeErdélyi 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) andKν⁡(z), including integrals with respect to the order, seeApelblat (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 (2015, §§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), andWheelon (1968).