Digital Library of Mathematical Functions
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10 Bessel FunctionsBessel and Hankel Functions

§10.16 Relations to Other Functions

Elementary Functions

10.16.1 J12(z) =Y-12(z)
=(2πz)12sinz,
J-12(z) =-Y12(z)
=(2πz)12cosz,
10.16.2 H12(1)(z) =-H-12(1)(z)
=-(2πz)12z,
H12(2)(z) =H-12(2)(z)
=(2πz)12-z.

For these and general results when ν is half an odd integer see §§10.47(ii) and 10.49(i).

Airy Functions

See §§9.6(i) and 9.6(ii).

Parabolic Cylinder Functions

With the notation of §12.14(i),

10.16.3 J14(z) =-2-14π-12z-14(W(0,2z12)-W(0,-2z12)),
J-14(z) =2-14π-12z-14(W(0,2z12)+W(0,-2z12)).
10.16.4 J34(z) =-2-14π-12z-34(W(0,2z12)-W(0,-2z12)),
J-34(z) =-2-14π-12z-34(W(0,2z12)+W(0,-2z12)).

Principal values on each side of these equations correspond.

Confluent Hypergeometric Functions

10.16.5 Jν(z)=(12z)νzΓ(ν+1)M(ν+12,2ν+1,±2z),
10.16.6 Hν(1)(z)Hν(2)(z)}=2π-12νπ(2z)ν±zU(ν+12,2ν+1,2z).

For the functions M and U see §13.2(i).

10.16.7 Jν(z)=(2ν+1)π/422νΓ(ν+1)(2z)-12M0,ν(±2z),
2ν-1,-2-3,,
10.16.8 Hν(1)(z)Hν(2)(z)}=(2ν+1)π/4(2πz)12W0,ν(2z).

For the functions M0,ν and W0,ν see §13.14(i).

In all cases principal branches correspond at least when |phz|12π.

Generalized Hypergeometric Functions

With F as in §15.2(i), and with z and ν fixed,

10.16.10 Jν(z)=(12z)νlimF(λ,μ;ν+1;-z2/(4λμ)),

as λ and μ in . For this result see Watson (1944, §5.7).