# §10.43 Integrals

## §10.43(i) Indefinite Integrals

Let be defined as in §10.25(ii). Then

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

10.43.3
,
.

## §10.43(ii) Integrals over the Intervals and

where and is Euler’s constant (§5.2).

## §10.43(iii) Fractional Integrals

The Bickley function is defined by

10.43.11

when and , and by analytic continuation elsewhere. Equivalently,

10.43.12.

### ¶ Properties

10.43.13
10.43.14
10.43.16.
10.43.17

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 ()

When ,

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

For the hypergeometric function see §15.2(i).

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 is defined as

10.43.30

Then

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

• (a)

On the interval , is continuously differentiable and each of and is absolutely integrable.

• (b)

is piecewise continuous and of bounded variation on every compact interval in , and each of the following integrals

• 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 and , 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).