Digital Library of Mathematical Functions
About the Project
NIST
10 Bessel FunctionsBessel and Hankel Functions

§10.19 Asymptotic Expansions for Large Order

Contents

§10.19(i) Asymptotic Forms

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

10.19.1 Jν(z)12πν(z2ν)ν,
10.19.2 Yν(z)-Hν(1)(z)Hν(2)(z)-2πν(z2ν)-ν.

§10.19(ii) Debye’s Expansions

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

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

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

10.19.5 ξ=ν(tanβ-β)-14π,

then

10.19.6 Jν(νsecβ) (2πνtanβ)12(cosξk=0U2k(cotβ)ν2k-sinξk=0U2k+1(cotβ)ν2k+1),
Yν(νsecβ) (2πνtanβ)12(sinξk=0U2k(cotβ)ν2k+cosξk=0U2k+1(cotβ)ν2k+1),
10.19.7 Jν(νsecβ) (sin(2β)πν)12(-sinξk=0V2k(cotβ)ν2k-cosξk=0V2k+1(cotβ)ν2k+1),
Yν(νsecβ) (sin(2β)πν)12(cosξk=0V2k(cotβ)ν2k-sinξk=0V2k+1(cotβ)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,
|phν|12π-δ,
Yν(ν+aν13) -213ν13Bi(-213a)k=0Pk(a)ν2k/3-223νBi(-213a)k=0Qk(a)ν2k/3,
|phν|12π-δ.

Also,

10.19.9 Hν(1)(ν+aν13)Hν(2)(ν+aν13)}243ν13π/3Ai(π/3213a)k=0Pk(a)ν2k/3+253ν±π/3Ai(π/3213a)k=0Qk(a)ν2k/3,

with sectors of validity -12π+δ±phν32π-δ. 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) =9577000a6-1733150a3-1225,
P4(a) =2720000a10-235731 47000a7+59031 38600a4+9473 46500a,
10.19.11 Q0(a) =310a2,
Q1(a) =-1770a3+170,
Q2(a) =-91000a7+6113150a4-373150a,
Q3(a) =-54928000a8-1 107676 93000a5+7912375a2.
10.19.12 Jν(ν+aν13) -223ν23Ai(-213a)k=0Rk(a)ν2k/3+213ν43Ai(-213a)k=0Sk(a)ν2k/3,
|phν|12π-δ,
Yν(ν+aν13) 223ν23Bi(-213a)k=0Rk(a)ν2k/3-213ν43Bi(-213a)k=0Sk(a)ν2k/3,
|phν|12π-δ.
10.19.13 Hν(1)(ν+aν13)Hν(2)(ν+aν13)}-253ν23±π/3Ai(π/3213a)k=0Rk(a)ν2k/3+243ν43π/3Ai(π/3213a)k=0Sk(a)ν2k/3,

with sectors of validity -12π+δphν32π-δ and -32π+δphν12π-δ, respectively. Here

10.19.14 R0(a) =1,
R1(a) =-45a,
R2(a) =-9100a5+5770a2,
R3(a) =6993500a6-26173150a3+233150,
R4(a) =2720000a10-466311 47000a7+38894620a4-11591 15500a,
10.19.15 S0(a) =35a3-15,
S1(a) =-131140a4+15a,
S2(a) =-9500a8+54374500a5-5933150a2,
S3(a) =3697000a9-9 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).