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11 Struve and Related FunctionsStruve and Modified Struve Functions

§11.6 Asymptotic Expansions

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§11.6(i) Large |z|, Fixed ν

11.6.1 Kν(z)1πk=0Γ(k+12)(12z)ν-2k-1Γ(ν+12-k),
|phz|π-δ,

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(zν-2m-1). If ν is real, z is positive, and m+12-ν0, 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)ν-2k-1Γ(ν+12-k),
|phz|12π-δ.

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)t-2π(ln(2z)+γ)2πk=1(-1)k+1(2k)!(2k-1)!(k!)2(2z)2k,
|phz|π-δ,
11.6.4 0zM0(t)t+2π(ln(2z)+γ)2πk=1(2k)!(2k-1)!(k!)2(2z)2k,
|phz|12π-δ,

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)ν-1πΓ(ν+12)k=0k!ck(λ)νk,
|phν|12π-δ,

and for fixed λ (>0)

11.6.7 Mν(λν)-(12λν)ν-1πΓ(ν+12)k=0k!ck(λ)νk,
|phν|12π-δ.

Here

11.6.8 c0(λ) =1,
c1(λ) =2λ-2,
c2(λ) =6λ-4-12λ-2,
c3(λ) =20λ-6-4λ-4,
c4(λ) =70λ-8-452λ-6+38λ-4,

and for higher coefficients ck(λ) see Dingle (1973, p. 203).

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),
|phν|12π-δ,

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