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10 Bessel FunctionsBessel and Hankel Functions

§10.19 Asymptotic Expansions for Large Order

  1. §10.19(i) Asymptotic Forms
  2. §10.19(ii) Debye’s Expansions
  3. §10.19(iii) Transition Region

§10.19(i) Asymptotic Forms

§10.19(ii) Debye’s Expansions

If ν through positive real values with α (>0) fixed, then

10.19.3 Jν(νsechα) eν(tanhαα)(2πνtanhα)12k=0Uk(cothα)νk,
Yν(νsechα) eν(αtanhα)(12πνtanhα)12k=0(1)kUk(cothα)νk,
10.19.4 Jν(νsechα) (sinh(2α)4πν)12eν(tanhαα)k=0Vk(cothα)νk,
Yν(νsechα) (sinh(2α)πν)12eν(αtanhα)k=0(1)kVk(cothα)νk.

If ν through positive real values with β ((0,12π)) fixed, and

10.19.5 ξ=ν(tanββ)14π,


10.19.6 Jν(νsecβ) (2πνtanβ)12(cosξk=0U2k(icotβ)ν2kisinξk=0U2k+1(icotβ)ν2k+1),
Yν(νsecβ) (2πνtanβ)12(sinξk=0U2k(icotβ)ν2k+icosξk=0U2k+1(icotβ)ν2k+1),
10.19.7 Jν(νsecβ) (sin(2β)πν)12(sinξk=0V2k(icotβ)ν2kicosξk=0V2k+1(icotβ)ν2k+1),
Yν(νsecβ) (sin(2β)πν)12(cosξk=0V2k(icotβ)ν2kisinξk=0V2k+1(icotβ)ν2k+1).

In these expansions Uk(p) and Vk(p) are the polynomials in p of degree 3k defined in §10.41(ii).

For error bounds for the first of (10.19.6) see Olver (1997b, p. 382).

§10.19(iii) Transition Region

As ν, with a() fixed,

10.19.8 Jν(ν+aν13) 213ν13Ai(213a)k=0Pk(a)ν2k/3+223νAi(213a)k=0Qk(a)ν2k/3,
Yν(ν+aν13) 213ν13Bi(213a)k=0Pk(a)ν2k/3223νBi(213a)k=0Qk(a)ν2k/3,


10.19.9 Hν(1)(ν+aν13)Hν(2)(ν+aν13)}243ν13eπi/3Ai(eπi/3213a)k=0Pk(a)ν2k/3+253νe±πi/3Ai(eπi/3213a)k=0Qk(a)ν2k/3,

with sectors of validity 12π+δ±phν32πδ. Here Ai and Bi are the Airy functions (§9.2), and

10.19.10 P0(a) =1,
P1(a) =15a,
P2(a) =9100a5+335a2,
P3(a) =9577000a61733150a31225,
P4(a) =2720000a10235731 47000a7+59031 38600a4+9473 46500a,
10.19.11 Q0(a) =310a2,
Q1(a) =1770a3+170,
Q2(a) =91000a7+6113150a4373150a,
Q3(a) =54928000a81 107676 93000a5+7912375a2.
10.19.12 Jν(ν+aν13) 223ν23Ai(213a)k=0Rk(a)ν2k/3+213ν43Ai(213a)k=0Sk(a)ν2k/3,
Yν(ν+aν13) 223ν23Bi(213a)k=0Rk(a)ν2k/3213ν43Bi(213a)k=0Sk(a)ν2k/3,
10.19.13 Hν(1)(ν+aν13)Hν(2)(ν+aν13)}253ν23e±πi/3Ai(eπi/3213a)k=0Rk(a)ν2k/3+243ν43eπi/3Ai(eπi/3213a)k=0Sk(a)ν2k/3,

with sectors of validity 12π+δphν32πδ and 32π+δphν12πδ, respectively. Here

10.19.14 R0(a) =1,
R1(a) =45a,
R2(a) =9100a5+5770a2,
R3(a) =6993500a626173150a3+233150,
R4(a) =2720000a10466311 47000a7+38894620a411591 15500a,
10.19.15 S0(a) =35a315,
S1(a) =131140a4+15a,
S2(a) =9500a8+54374500a55933150a2,
S3(a) =3697000a99 994436 93000a6+317271 73250a3+9473 46500.

For proofs and also for the corresponding expansions for second derivatives see Olver (1952).

For higher coefficients in (10.19.8) in the case a=0 (that is, in the expansions of Jν(ν) and Yν(ν)), see Watson (1944, §8.21), Temme (1997), and Jentschura and Lötstedt (2012). The last reference also includes the corresponding expansions for Jν(ν) and Yν(ν).

See also §10.20(i).