§10.22 Integrals
Contents
- §10.22(i) Indefinite Integrals
- §10.22(ii) Integrals over Finite Intervals
- §10.22(iii) Integrals over the Interval
- §10.22(iv) Integrals over the Interval
- §10.22(v) Hankel Transform
- §10.22(vi) Compendia
§10.22(i) Indefinite Integrals
In this subsection and denote cylinder functions(§10.2(ii)) of orders and , respectively, not necessarily distinct.
For the Struve function see §11.2(i).
¶ Products
§10.22(ii) Integrals over Finite Intervals
Throughout this subsection .
¶ Trigonometric Arguments
For see §10.25(ii).
¶ Products
¶ Convolutions
¶ Fractional Integral
§10.22(iii) Integrals over the Interval
§10.22(iv) Integrals over the Interval
For see §10.25(ii).
For the hypergeometric function see §15.2(i).
For and see §10.25(ii).
For the confluent hypergeometric function see §13.2(i).
¶ Orthogonality
¶ Weber–Schafheitlin Discontinuous Integrals, including Special Cases
If , then interchange and , and also and . If , then
When
When ,
When ,
When ,
When and ,
¶ 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 (10.22.71) and (10.22.72) are positive constants.
For the Ferrers function and the associated Legendre function , see §§14.3(i) and 14.3(ii), respectively.
In (10.22.74) and (10.22.75), are positive constants and
(Thus if are the sides of a triangle, then is the area of the triangle.)
If , then
If , then
§10.22(v) Hankel Transform
The Hankel transform (or Bessel transform) of a function is defined as
§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).