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10 Bessel FunctionsBessel and Hankel Functions

§10.17 Asymptotic Expansions for Large Argument

Contents

§10.17(i) Hankel’s Expansions

Define a0(ν)=1,

10.17.1 ak(ν)=(4ν2-12)(4ν2-32)(4ν2-(2k-1)2)k!8k,
k1,
10.17.2 ω=z-12νπ-14π,

and let δ denote an arbitrary small positive constant. Then as z, with ν fixed,

10.17.3 Jν(z)(2πz)12(cosωk=0(-1)ka2k(ν)z2k-sinωk=0(-1)ka2k+1(ν)z2k+1),
|phz|π-δ,
10.17.4 Yν(z)(2πz)12(sinωk=0(-1)ka2k(ν)z2k+cosωk=0(-1)ka2k+1(ν)z2k+1),
|phz|π-δ,
10.17.5 Hν(1)(z)(2πz)12eiωk=0ikak(ν)zk,
-π+δphz2π-δ,
10.17.6 Hν(2)(z)(2πz)12e-iωk=0(-i)kak(ν)zk,
-2π+δphzπ-δ,

where the branch of z12 is determined by

10.17.7 z12=exp(12ln|z|+12iphz).

Corresponding expansions for other ranges of phz can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4).

§10.17(ii) Asymptotic Expansions of Derivatives

We continue to use the notation of §10.17(i). Also, b0(ν)=1, b1(ν)=(4ν2+3)/8, and for k2,

10.17.8 bk(ν)=((4ν2-12)(4ν2-32)(4ν2-(2k-3)2))(4ν2+4k2-1)k!8k.

Then as z with ν fixed,

10.17.9 Jν(z) -(2πz)12(sinωk=0(-1)kb2k(ν)z2k+cosωk=0(-1)kb2k+1(ν)z2k+1),
|phz|π-δ,
10.17.10 Yν(z) (2πz)12(cosωk=0(-1)kb2k(ν)z2k-sinωk=0(-1)kb2k+1(ν)z2k+1),
|phz|π-δ,
10.17.11 Hν(1)(z) i(2πz)12eiωk=0ikbk(ν)zk,
-π+δphz2π-δ,
10.17.12 Hν(2)(z) -i(2πz)12e-iωk=0(-i)kbk(ν)zk,
-2π+δphzπ-δ.

§10.17(iii) Error Bounds for Real Argument and Order

In the expansions (10.17.3) and (10.17.4) assume that ν0 and z>0. Then the remainder associated with the sum k=0-1(-1)ka2k(ν)z-2k does not exceed the first neglected term in absolute value and has the same sign provided that max(12ν-14,1). Similarly for k=0-1(-1)ka2k+1(ν)z-2k-1, provided that max(12ν-34,1).

In the expansions (10.17.5) and (10.17.6) assume that ν>-12 and z>0. If these expansions are terminated when k=-1, then the remainder term is bounded in absolute value by the first neglected term, provided that max(ν-12,1).

§10.17(iv) Error Bounds for Complex Argument and Order

For (10.17.5) and (10.17.6) write

10.17.13 Hν(1)(z)Hν(2)(z)}=(2πz)12e±iω(k=0-1(±i)kak(ν)zk+R±(ν,z)),
=1,2,.

Then

10.17.14 |R±(ν,z)|2|a(ν)|𝒱z,±i(t-)exp(|ν2-14|𝒱z,±i(t-)),

where 𝒱 denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that |t| changes monotonically. Bounds for 𝒱z,i(t-) are given by

10.17.15 𝒱z,i(t-){|z|-,0phzπ,χ()|z|-,-12πphz0 or πphz32π,2χ()|z|-,-π<phz-12π or 32πphz<2π,

where χ()=π12Γ(12+1)/Γ(12+12); see §9.7(i). The bounds (10.17.15) also apply to 𝒱z,-i(t-) in the conjugate sectors.

Corresponding error bounds for (10.17.3) and (10.17.4) are obtainable by combining (10.17.13) and (10.17.14) with (10.4.4).

§10.17(v) Exponentially-Improved Expansions

As in §9.7(v) denote

10.17.16 Gp(z)=ez2πΓ(p)Γ(1-p,z),

where Γ(1-p,z) is the incomplete gamma function (§8.2(i)). Then in (10.17.13) as z with |-2|z|| bounded and m (0) fixed,

10.17.17 R±(ν,z)=(-1)2cos(νπ)(k=0m-1(±i)kak(ν)zkG-k(2iz)+Rm,±(ν,z)),

where

10.17.18 Rm,±(ν,z)=O(e-2|z|z-m),
|ph(ze12πi)|π.

For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).