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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.7 Integrals and Sums

Contents
  1. §11.7(i) Indefinite Integrals
  2. §11.7(ii) Definite Integrals
  3. §11.7(iii) Laplace Transforms
  4. §11.7(iv) Integrals with Respect to Order
  5. §11.7(v) Compendia

§11.7(i) Indefinite Integrals

11.7.1 zν𝐇ν1(z)dz=zν𝐇ν(z),
11.7.2 zν𝐇ν+1(z)dz=zν𝐇ν(z)+2νzπΓ(ν+32),
11.7.3 zν𝐋ν1(z)dz=zν𝐋ν(z),
11.7.4 zν𝐋ν+1(z)dz=zν𝐋ν(z)2νzπΓ(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=2νπΓ(12ν)zν1(1cosz),
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 0tν1𝐇ν(t)dt=π2ν+1Γ(ν+1),
ν>32,
11.7.11 0tμν1𝐇ν(t)dt=Γ(12μ)2μν1tan(12πμ)Γ(ν12μ+1),
|μ|<1, ν>μ32,
11.7.12 0tμν𝐇μ(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 0eat𝐇0(t)dt=2π1+a2ln(1+1+a2a),
11.7.14 0eat𝐇1(t)dt=2πa2aπ1+a2ln(1+1+a2a),
11.7.15 0eat𝐋0(t)dt=2πa21arcsin(1a),
11.7.16 0eat𝐋1(t)dt
=2aπa21arctan(1a21)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). See also Baricz and Pogány (2013)