Digital Library of Mathematical Functions
About the Project
NIST
11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.5 Integral Representations

Contents

§11.5(i) Integrals Along the Real Line

11.5.1 Hν(z)=2(12z)νπΓ(ν+12)01(1-t2)ν-12sin(zt)t=2(12z)νπΓ(ν+12)0π/2sin(zcosθ)(sinθ)2νθ,
ν>-12,
11.5.2 Kν(z)=2(12z)νπΓ(ν+12)0-zt(1+t2)ν-12t,
z>0,
11.5.3 K0(z)=2π0-zsinhtt,
z>0,
11.5.4 Mν(z)=-2(12z)νπΓ(ν+12)01-zt(1-t2)ν-12t,
ν>-12,
11.5.5 M0(z)=-2π0π/2-zcosθθ,
11.5.6 Lν(z)=2(12z)νπΓ(ν+12)0π/2sinh(zcosθ)(sinθ)2νθ,
ν>-12,
11.5.7 I-ν(x)-Lν(x)=2(12x)νπΓ(ν+12)0(1+t2)ν-12sin(xt)t,
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)-ν-1Hν(x)=-12π-πcsc(πs)Γ(32+s)Γ(32+ν+s)(14x2)ss,
x>0, ν>-1,
11.5.9 (12z)-ν-1Lν(z)=12π(0+)πcsc(πs)Γ(32+s)Γ(32+ν+s)(-14z2)ss.

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).