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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.5 Integral Representations

Contents
  1. §11.5(i) Integrals Along the Real Line
  2. §11.5(ii) Contour Integrals
  3. §11.5(iii) Compendia

§11.5(i) Integrals Along the Real Line

11.5.1 𝐇ν(z)=2(12z)νπΓ(ν+12)01(1t2)ν12sin(zt)dt=2(12z)νπΓ(ν+12)0π/2sin(zcosθ)(sinθ)2νdθ,
ν>12,
11.5.2 𝐊ν(z)=2(12z)νπΓ(ν+12)0ezt(1+t2)ν12dt,
z>0,
11.5.3 𝐊0(z)=2π0ezsinhtdt,
z>0,
11.5.4 𝐌ν(z)=2(12z)νπΓ(ν+12)01ezt(1t2)ν12dt,
ν>12,
11.5.5 𝐌0(z)=2π0π/2ezcosθdθ,
11.5.6 𝐋ν(z)=2(12z)νπΓ(ν+12)0π/2sinh(zcosθ)(sinθ)2νdθ,
ν>12,
11.5.7 Iν(x)𝐋ν(x)=2(12x)νπΓ(ν+12)0(1+t2)ν12sin(xt)dt,
x>0, ν<12.

§11.5(ii) Contour Integrals

For loop-integral versions of (11.5.1), (11.5.2), (11.5.4), and (11.5.7) see Babister (1967, §§3.3 and 3.14).

Mellin–Barnes Integrals

11.5.8 (12x)ν1𝐇ν(x)=12πiiiπcsc(πs)Γ(32+s)Γ(32+ν+s)(14x2)sds,
x>0, ν>1,
11.5.9 (12z)ν1𝐋ν(z)=12πi(0+)πcsc(πs)Γ(32+s)Γ(32+ν+s)(14z2)sds.

In (11.5.8) and (11.5.9) the path of integration separates the poles of the integrand at s=0,1,2, from those at s=1,2,3,.

§11.5(iii) Compendia

For further integral representations see Babister (1967, §§3.3, 3.14), Erdélyi et al. (1954a, §§5.17, 15.3), Magnus et al. (1966, p. 114), Oberhettinger (1972), Oberhettinger (1974, §2.7), Oberhettinger and Badii (1973, §2.14), and Watson (1944, pp. 330, 374, and 426).