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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.6 Asymptotic Expansions

  1. §11.6(i) Large |z|, Fixed ν
  2. §11.6(ii) Large |ν|, Fixed z
  3. §11.6(iii) Large |ν|, Fixed z/ν

§11.6(i) Large |z|, Fixed ν

11.6.1 𝐊ν(z)1πk=0Γ(k+12)(12z)ν2k1Γ(ν+12k),

where δ is an arbitrary small positive constant. If the series on the right-hand side of (11.6.1) is truncated after m(0) terms, then the remainder term Rm(z) is O(zν2m1). If ν is real, z is positive, and m+12ν0, then Rm(z) is of the same sign and numerically less than the first neglected term.

11.6.2 𝐌ν(z)1πk=0(1)k+1Γ(k+12)(12z)ν2k1Γ(ν+12k),

For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445).

For the corresponding expansions for 𝐇ν(z) and 𝐋ν(z) combine (11.6.1), (11.6.2) with (11.2.5), (11.2.6), (10.17.4), and (10.40.1).

11.6.3 0z𝐊0(t)dt2π(ln(2z)+γ)2πk=1(1)k+1(2k)!(2k1)!(k!)2(2z)2k,
11.6.4 0z𝐌0(t)dt+2π(ln(2z)+γ)2πk=1(2k)!(2k1)!(k!)2(2z)2k,

where γ is Euler’s constant (§5.2(ii)).

§11.6(ii) Large |ν|, Fixed z

More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)).

§11.6(iii) Large |ν|, Fixed z/ν

For fixed λ(>1)

11.6.6 𝐊ν(λν)(12λν)ν1πΓ(ν+12)k=0k!ck(λ)νk,

and for fixed λ (>0)

11.6.7 𝐌ν(λν)(12λν)ν1πΓ(ν+12)k=0k!ck(iλ)νk,


11.6.8 c0(λ) =1,
c1(λ) =2λ2,
c2(λ) =6λ412λ2,
c3(λ) =20λ64λ4,
c4(λ) =70λ8452λ6+38λ4.

These and higher coefficients ck(λ) can be computed via the representations in Nemes (2015b).

For the corresponding result for 𝐇ν(λν) use (11.2.5) and (10.19.6). See also Watson (1944, p. 336).

For fixed λ (>0)

11.6.9 𝐋ν(λν)Iν(λν),

and for an estimate of the relative error in this approximation see Watson (1944, p. 336).