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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.23 Integrals

Contents
  1. §13.23(i) Laplace and Mellin Transforms
  2. §13.23(ii) Fourier Transforms
  3. §13.23(iii) Hankel Transforms
  4. §13.23(iv) Integral Transforms in terms of Whittaker Functions
  5. §13.23(v) Other Integrals

§13.23(i) Laplace and Mellin Transforms

For the notation see §§15.1, 15.2(i), and 10.25(ii).

13.23.1 0ezttν1Mκ,μ(t)dt=Γ(μ+ν+12)(z+12)μ+ν+12F12(12+μκ,12+μ+ν1+2μ;1z+12),
μ+ν+12>0, z>12.
13.23.2 0ezttμ12Mκ,μ(t)dt=Γ(2μ+1)(z+12)κμ12(z12)κμ12,
μ>12, z>12,
13.23.3 1Γ(1+2μ)0e12ttν1Mκ,μ(t)dt=Γ(μ+ν+12)Γ(κν)Γ(12+μ+κ)Γ(12+μν),
12μ<ν<κ.
13.23.4 0ezttν1Wκ,μ(t)dt=Γ(12+μ+ν)Γ(12μ+ν)𝐅12(12μ+ν,12+μ+ννκ+1;12z),
(ν+12)>|μ|, z>12,
13.23.5 0e12ttν1Wκ,μ(t)dt=Γ(12+μ+ν)Γ(12μ+ν)Γ(κν)Γ(12+μκ)Γ(12μκ),
|μ|12<ν<κ.
13.23.6 1Γ(1+2μ)2πi(0+)ezt+12t1tκMκ,μ(t1)dt=zκ12Γ(12+μκ)I2μ(2z),
z>0.
13.23.7 12πi(0+)ezt+12t1tκWκ,μ(t1)dt=2zκ12Γ(12+μκ)Γ(12μκ)K2μ(2z),
z>0.

For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), and Prudnikov et al. (1992a, §§3.34, 3.35). Inverse Laplace transforms are given in Oberhettinger and Badii (1973, §2.16) and Prudnikov et al. (1992b, §§3.33, 3.34).

§13.23(ii) Fourier Transforms

13.23.8 1Γ(1+2μ)0cos(2xt)e12t2t2μ1Mκ,μ(t2)dt=πe12x2xμ+κ12Γ(12+μ+κ)W12κ32μ,12κ+12μ(x2),
(κ+μ)>12.

For additional Fourier transforms see Erdélyi et al. (1954a, §§1.14, 2.14, 3.3) and Oberhettinger (1990, §§1.22, 2.22).

§13.23(iii) Hankel Transforms

For the notation see §10.2(ii).

13.23.9 0e12ttμ12(ν+1)Mκ,μ(t)Jν(2xt)dt=Γ(1+2μ)Γ(12μ+κ+ν)e12xx12(κμ32)M12(κ+3μν+12),12(κμ+ν12)(x),
x>0, 12<μ<(κ+12ν)+34,
13.23.10 1Γ(1+2μ)0e12tt12(ν1)μMκ,μ(t)Jν(2xt)dt=e12xx12(κ+μ32)Γ(12+μ+κ)W12(κ3μ+ν+12),12(κ+μν12)(x),
x>0, 1<ν<2(μ+κ)+12.
13.23.11 0e12tt12(ν1)μWκ,μ(t)Jν(2xt)dt=Γ(ν2μ+1)Γ(12+μκ)e12xx12(μκ32)W12(κ+3μν12),12(κμ+ν+12)(x),
x>0, max(2μ1,1)<ν<2μκ+32,
13.23.12 0e12tt12(ν1)μWκ,μ(t)Jν(2xt)dt=Γ(ν2μ+1)Γ(32μκ+ν)e12xx12(μ+κ32)M12(κ3μ+ν+12),12(νμκ+12)(x),
x>0, max(2μ1,1)<ν.

For additional Hankel transforms and also other Bessel transforms see Erdélyi et al. (1954b, §8.18) and Oberhettinger (1972, §1.16 and 3.4.42–46, 4.4.45–47, 5.94–97).

§13.23(iv) Integral Transforms in terms of Whittaker Functions

Let f(x) be absolutely integrable on the interval [r,R] for all positive r<R, f(x)=O(xρ0) as x0+, and f(x)=O(eρ1x) as x+, where ρ1>12. Then for μ in the half-plane μμ1>max(ρ0,κ12)

13.23.13 g(μ) =1Γ(1+2μ)0f(x)x32Mκ,μ(x)dx,
13.23.14 f(x) =1πixμ1iμ1+iμg(μ)Γ(12+μκ)Wκ,μ(x)dμ.

For additional integral transforms see Magnus et al. (1966, p. 189), Prudnikov et al. (1992b, §§4.3.39–4.3.42), and Wimp (1964).

§13.23(v) Other Integrals

Additional integrals involving confluent hypergeometric functions can be found in Apelblat (1983, pp. 388–392), Erdélyi et al. (1954b), Gradshteyn and Ryzhik (2000, §7.6), and Prudnikov et al. (1990, §§1.13, 1.14, 2.19, 4.2.2). See also (13.16.2), (13.16.6), (13.16.7). Generalized orthogonality integrals (33.14.13) and (33.14.15) can be expressed in terms of Whittaker functions via the definitions in that section.