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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.4 Basic Properties

  1. §11.4(i) Half-Integer Orders
  2. §11.4(ii) Inequalities
  3. §11.4(iii) Analytic Continuation
  4. §11.4(iv) Expansions in Series of Bessel Functions
  5. §11.4(v) Recurrence Relations and Derivatives
  6. §11.4(vi) Derivatives with Respect to Order
  7. §11.4(vii) Zeros

§11.4(i) Half-Integer Orders

For n=0,1,2,,

11.4.1 𝐊n+12(z)=(2πz)12m=0n(2m)! 22mm!(nm)!(12z)n2m,
11.4.2 𝐋n+12(z)=In12(z)(2πz)12m=0n(1)m(2m)! 22mm!(nm)!(12z)n2m,
11.4.3 𝐇n12(z) =(1)nJn+12(z),
11.4.4 𝐋n12(z) =In+12(z).
11.4.5 𝐇12(z) =(2πz)12(1cosz),
11.4.6 𝐇12(z) =(2πz)12sinz,
11.4.7 𝐋12(z) =(2πz)12(coshz1),
11.4.8 𝐋12(z) =(2πz)12sinhz,
11.4.9 𝐇32(z)=(z2π)12(1+2z2)(2πz)12(sinz+coszz),
11.4.10 𝐇32(z)=(2πz)12(coszsinzz),
11.4.11 𝐋32(z)=(z2π)12(12z2)+(2πz)12(sinhzcoshzz),
11.4.12 𝐋32(z)=(2πz)12(coshzsinhzz).

§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),


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(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),
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(1J0(z))+4πk=1J2k(z)4k21=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 𝐇ν1(z)+𝐇ν+1(z) =2νz𝐇ν(z)+(12z)νπΓ(ν+32),
11.4.24 𝐇ν1(z)𝐇ν+1(z) =2𝐇ν(z)(12z)νπΓ(ν+32),
11.4.25 𝐋ν1(z)𝐋ν+1(z) =2νz𝐋ν(z)+(12z)νπΓ(ν+32),
11.4.26 𝐋ν1(z)+𝐋ν+1(z) =2𝐋ν(z)(12z)νπΓ(ν+32).
11.4.27 ddz(zν𝐇ν(z))=zν𝐇ν1(z),
11.4.28 ddz(zν𝐇ν(z))=2νπΓ(ν+32)zν𝐇ν+1(z),
11.4.29 ddz(zν𝐋ν(z))=zν𝐋ν1(z),
11.4.30 ddz(zν𝐋ν(z))=2νπΓ(ν+32)+zν𝐋ν+1(z).
11.4.31 νm(z)=zmν(1zddz)m(zνν(z)),

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

11.4.32 𝐇0(z) =2π𝐇1(z),
ddz(z𝐇1(z)) =z𝐇0(z),
11.4.33 𝐋0(z) =2π+𝐋1(z),
ddz(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).