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10 Bessel FunctionsBessel and Hankel Functions

§10.7 Limiting Forms

  1. §10.7(i) z0
  2. §10.7(ii) z

§10.7(i) z0

When ν is fixed and z0,

10.7.1 J0(z) 1,
Y0(z) (2/π)lnz,
10.7.2 H0(1)(z)H0(2)(z)(2i/π)lnz,
10.7.3 Jν(z)(12z)ν/Γ(ν+1),
10.7.4 Yν(z) (1/π)Γ(ν)(12z)ν,
ν>0 or ν=12,32,52,,
10.7.5 Yν(z) (1/π)cos(νπ)Γ(ν)(12z)ν,
ν>0, ν12,32,52,,
10.7.6 Yiν(z)=icsch(νπ)Γ(1iν)(12z)iνicoth(νπ)Γ(1+iν)(12z)iν+e|νphz|o(1),
ν and ν0.

See also §10.24 when z=x (>0).

10.7.7 Hν(1)(z)Hν(2)(z)(i/π)Γ(ν)(12z)ν,

For Hν(1)(z) and Hν(2)(z) when ν>0 combine (10.4.6) and (10.7.7). For Hiν(1)(z) and Hiν(2)(z) when ν and ν0 combine (10.4.3), (10.7.3), and (10.7.6).

§10.7(ii) z

When ν is fixed and z,

10.7.8 Jν(z) =2/(πz)(cos(z12νπ14π)+e|z|o(1)),
Yν(z) =2/(πz)(sin(z12νπ14π)+e|z|o(1)),

For the corresponding results for Hν(1)(z) and Hν(2)(z) see (10.2.5) and (10.2.6).