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10 Bessel FunctionsModified Bessel Functions

§10.40 Asymptotic Expansions for Large Argument

  1. §10.40(i) Hankel’s Expansions
  2. §10.40(ii) Error Bounds for Real Argument and Order
  3. §10.40(iii) Error Bounds for Complex Argument and Order
  4. §10.40(iv) Exponentially-Improved Expansions

§10.40(i) Hankel’s Expansions

With the notation of §§10.17(i) and 10.17(ii), as z with ν fixed,

10.40.1 Iν(z) ez(2πz)12k=0(1)kak(ν)zk,
10.40.2 Kν(z) (π2z)12ezk=0ak(ν)zk,
10.40.3 Iν(z)ez(2πz)12k=0(1)kbk(ν)zk,
10.40.4 Kν(z)(π2z)12ezk=0bk(ν)zk,

Corresponding expansions for Iν(z), Kν(z), Iν(z), and Kν(z) for other ranges of phz are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). In particular, use of (10.34.3) with m=0 yields the following more general (and more accurate) version of (10.40.1):

10.40.5 Iν(z)ez(2πz)12k=0(1)kak(ν)zk±ie±νπiez(2πz)12k=0ak(ν)zk,


With μ=4ν2 and fixed,

10.40.6 Iν(z)Kν(z) 12z(112μ1(2z)2+1324(μ1)(μ9)(2z)4),
10.40.7 Iν(z)Kν(z) 12z(1+12μ3(2z)2124(μ1)(μ45)(2z)4+),

as z in |phz|12πδ. The general terms in (10.40.6) and (10.40.7) can be written down by analogy with (10.18.17), (10.18.19), and (10.18.20).


For fixed ν,

10.40.8 Kν(z)ν(π2z)12νezzk=0αk(ν)(8z)k,

as z in |phz|32πδ. Here α0(ν)=1 and

10.40.9 αk(ν)=(4ν212)(4ν232)(4ν2(2k+1)2)(k+1)!×(14ν212+14ν232++14ν2(2k+1)2).

§10.40(ii) Error Bounds for Real Argument and Order

In the expansion (10.40.2) assume that z>0 and the sum is truncated when k=1. Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that max(|ν|12,1).

For the error term in (10.40.1) see §10.40(iii).

§10.40(iii) Error Bounds for Complex Argument and Order

For (10.40.2) write

10.40.10 Kν(z)=(π2z)12ez(k=01ak(ν)zk+R(ν,z)),


10.40.11 |R(ν,z)|2|a(ν)|𝒱z,(t)exp(|ν214|𝒱z,(t1)),

where 𝒱 denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that |t| changes monotonically. Bounds for 𝒱z,(t) are given by

10.40.12 𝒱z,(t){|z|,|phz|12π,χ()|z|,12π|phz|π,2χ()|z|,π|phz|<32π,

where χ()=π12Γ(12+1)/Γ(12+12); see §9.7(i).

A similar result for (10.40.1) is obtained by combining (10.34.3), with m=0, and (10.40.10)–(10.40.12); see Olver (1997b, p. 269).

§10.40(iv) Exponentially-Improved Expansions

In (10.40.10)

10.40.13 R(ν,z)=(1)2cos(νπ)(k=0m1ak(ν)zkGk(2z)+Rm,(ν,z)),

where Gp(z) is given by (10.17.16). If z with |2|z|| bounded and m (0) fixed, then

10.40.14 Rm,(ν,z)=O(e2|z|zm),

For higher re-expansions of the remainder term see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Paris (2001a, b).