Digital Library of Mathematical Functions
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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.4 Basic Properties

Contents

§11.4(i) Half-Integer Orders

For n=0,1,2,,

11.4.1 Kn+12(z)=(2πz)12m=0n(2m)! 2-2mm!(n-m)!(12z)n-2m,
11.4.2 Ln+12(z)=I-n-12(z)-(2πz)12m=0n(-1)m(2m)! 2-2mm!(n-m)!(12z)n-2m,
11.4.3 H-n-12(z) =(-1)nJn+12(z),
11.4.4 L-n-12(z) =In+12(z).
11.4.5 H12(z) =(2πz)12(1-cosz),
11.4.6 H-12(z) =(2πz)12sinz,
11.4.7 L12(z) =(2πz)12(coshz-1),
11.4.8 L-12(z) =(2πz)12sinhz,
11.4.9 H32(z)=(z2π)12(1+2z2)-(2πz)12(sinz+coszz),
11.4.10 H-32(z)=(2πz)12(cosz-sinzz),
11.4.11 L32(z)=-(z2π)12(1-2z2)+(2πz)12(sinhz-coshzz),
11.4.12 L-32(z)=(2πz)12(coshz-sinhzz).

§11.4(ii) Inequalities

11.4.13 Hν(x)0,
x>0, ν12.
11.4.14 Hν(z)=2(12z)ν+1πΓ(ν+32)(1+ϑ),
ν-32,-52,-72,,

where

11.4.15 |ϑ|<23exp(14|z|2|ν0+32|-1),

and |ν0+32| is the smallest of the numbers |ν+32|, |ν+52|, |ν+92|,.

§11.4(iii) Analytic Continuation

11.4.16 Hν(zmπ)=mπ(ν+1)Hν(z),
m,
11.4.17 Lν(zmπ)=mπ(ν+1)Lν(z),
m.

§11.4(iv) Expansions in Series of Bessel Functions

11.4.18 Hν(z)=4π1/2Γ(ν+12)k=0(2k+ν+1)Γ(k+ν+1)k!(2k+1)(2k+2ν+1)J2k+ν+1(z),
ν-1,-2,-3,,
11.4.19 Hν(z)=(z2π)1/2k=0(12z)kk!(k+12)Jk+ν+12(z),
11.4.20 Hν(z)=(12z)ν+12Γ(ν+12)k=0(12z)kk!(k+ν+12)Jk+12(z),
11.4.21 H0(z)=4πk=0J2k+1(z)2k+1=2k=0(-1)kJk+122(12z),
11.4.22 H1(z)=2π(1-J0(z))+4πk=1J2k(z)4k2-1=4k=0J2k+12(12z)J2k+32(12z).

For these and further results see Luke (1969b, §9.4.5), and §10.23(iii).

§11.4(v) Recurrence Relations and Derivatives

11.4.23 Hν-1(z)+Hν+1(z) =2νzHν(z)+(12z)νπΓ(ν+32),
11.4.24 Hν-1(z)-Hν+1(z) =2Hν(z)-(12z)νπΓ(ν+32),
11.4.25 Lν-1(z)-Lν+1(z) =2νzLν(z)+(12z)νπΓ(ν+32),
11.4.26 Lν-1(z)+Lν+1(z) =2Lν(z)-(12z)νπΓ(ν+32).
11.4.27 z(zνHν(z))=zνHν-1(z),
11.4.28 z(z-νHν(z))=2-νπΓ(ν+32)-z-νHν+1(z),
11.4.29 z(zνLν(z))=zνLν-1(z),
11.4.30 z(z-νLν(z))=2-νπΓ(ν+32)+z-νLν+1(z).
11.4.31 ν-m(z)=zm-ν(1zz)m(zνν(z)),
m=1,2,3,,

where ν(z) denotes either Hν(z) or Lν(z).

11.4.32 H0(z) =2π-H1(z),
z(zH1(z)) =zH0(z),
11.4.33 L0(z) =2π+L1(z),
z(zL1(z)) =zL0(z).

§11.4(vi) Derivatives with Respect to Order

For derivatives with respect to the order ν, see Apelblat (1989) and Brychkov and Geddes (2005).

§11.4(vii) Zeros

For properties of zeros of Hν(x) see Steinig (1970).

For asymptotic expansions of zeros of H0(x) see MacLeod (2002a).