§10.22 Integrals

§10.22(i) Indefinite Integrals

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

10.22.1

For the Struve function see §11.2(i).

§10.22(ii) Integrals over Finite Intervals

Throughout this subsection .

¶ Trigonometric Arguments

For see §10.25(ii).

¶ Fractional Integral

When the left-hand side of (10.22.36) is the th repeated integral of (§§1.4(v) and 1.15(vi)).

¶ Orthogonality

If , then

where and are zeros of 10.21(i)), and is Kronecker’s symbol.

Also, if are real constants with and , then

where and are positive zeros of . (Compare (10.22.55)).

§10.22(iv) Integrals over the Interval

10.22.45.
10.22.47.

For see §10.25(ii).

10.22.50.

For the hypergeometric function see §15.2(i).

For and see §10.25(ii).

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

When ,

10.22.63

¶ Other Double Products

In (10.22.66)–(10.22.70) are positive constants.

For the associated Legendre function see §14.3(ii) with . For and see §10.25(ii).

Equation (10.22.70) also remains valid if the order of the functions on both sides is replaced by , , and the constraint is replaced by .

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

¶ Triple Products

In (10.22.74) and (10.22.75), are positive constants and

10.22.73

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

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 is defined as

Hankel’s inversion theorem is given by

Sufficient conditions for the validity of (10.22.77) are that when , or that and when ; 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 , , , and , 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).