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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.7 Integrals and Sums

Contents

§11.7(i) Indefinite Integrals

11.7.1 zνHν-1(z)dz=zνHν(z),
11.7.2 z-νHν+1(z)dz=-z-νHν(z)+2-νzπΓ(ν+32),
11.7.3 zνLν-1(z)dz=zνLν(z),
11.7.4 z-νLν+1(z)dz=z-νLν(z)-2-νzπΓ(ν+32).

If

11.7.5 fν(z)=0ztνHν(t)dt,

then

11.7.6 fν+1(z)=(2ν+1)fν(z)-zν+1Hν(z)+(12z2)ν+1(ν+1)πΓ(ν+32),
ν>-1.

§11.7(ii) Definite Integrals

11.7.7 0π/2Hν(zsinθ)(sinθ)ν+1(cosθ)2νdθ=2-νπΓ(12-ν)zν-1(1-cosz),
-32<ν<12,
11.7.8 0H0(t)dtt =12π,
0H1(t)dtt2 =14π,
11.7.9 0Hν(t)dt=-cot(12πν),
-2<ν<0,
11.7.10 0t-ν-1Hν(t)dt=π2ν+1Γ(ν+1),
ν>-32,
11.7.11 0tμ-ν-1Hν(t)dt=Γ(12μ)2μ-ν-1tan(12πμ)Γ(ν-12μ+1),
|μ|<1, ν>μ-32,
11.7.12 0t-μ-νHμ(t)Hν(t)dt=πΓ(μ+ν)2μ+νΓ(μ+ν+12)Γ(μ+12)Γ(ν+12),
(μ+ν)>0.

For other integrals involving products of Struve functions see Zanovello (1978, 1995). For integrals involving products of Mν(t) functions, see Paris and Sy (1983, Appendix).

§11.7(iii) Laplace Transforms

The following Laplace transforms of Hν(t) require a>0 for convergence, while those of Lν(t) require a>1.

11.7.13 0e-atH0(t)dt=2π1+a2ln(1+1+a2a),
11.7.14 0e-atH1(t)dt=2πa-2aπ1+a2ln(1+1+a2a),
11.7.15 0e-atL0(t)dt=2πa2-1arcsin(1a),
11.7.16 0e-atL1(t)dt
=2aπa2-1arctan(1a2-1)-2πa.

§11.7(iv) Integrals with Respect to Order

For integrals of Hν(x) and Lν(x) with respect to the order ν, see Apelblat (1989).

§11.7(v) Compendia

For further integrals see Apelblat (1983, §12.16), Babister (1967, Chapter 3), Erdélyi et al. (1954a, §§4.19, 6.8, 8.15, 9.4, 10.3, 11.3, and 15.3), Luke (1962, Chapters 9, 11), Gradshteyn and Ryzhik (2000, §6.8), Marichev (1983, pp. 192–193 and 215–216), Oberhettinger (1972), Oberhettinger (1974, §1.12), Oberhettinger (1990, §§1.21 and 2.21), Oberhettinger and Badii (1973, §1.16), Prudnikov et al. (1990, §§1.4 and 2.7), Prudnikov et al. (1992a, §3.17), and Prudnikov et al. (1992b, §3.17).

For sums of Struve functions see Hansen (1975, p. 456) and Prudnikov et al. (1990, §6.4.1).