The Anger function and Weber function are defined by
Each is an entire function of and . Also,
The associated Anger–Weber function is defined by
(11.10.4) also applies when and .
The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation
These expansions converge absolutely for all finite values of .
For the Fresnel integrals and see §7.2(iii).
where the prime on the second summation symbols means that the first term is to be halved.
For collections of integral representations and integrals see Erdélyi et al. (1954a, §§4.19 and 5.17), Marichev (1983, pp. 194–195 and 214–215), Oberhettinger (1972, p. 128), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105 and 189–190), Prudnikov et al. (1990, §§1.5 and 2.8), Prudnikov et al. (1992a, §3.18), Prudnikov et al. (1992b, §3.18), and Zanovello (1977).