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10 Bessel FunctionsModified Bessel Functions

§10.39 Relations to Other Functions

Elementary Functions

10.39.1 I12(z) =(2πz)12sinhz,
I-12(z) =(2πz)12coshz,
10.39.2 K12(z)=K-12(z)=(π2z)12e-z.

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

Airy Functions

See §§9.6(i) and 9.6(ii).

Parabolic Cylinder Functions

With the notation of §12.2(i),

10.39.3 K14(z)=π12z-14U(0,2z12),
10.39.4 K34(z)=12π12z-34(12U(1,2z12)+U(-1,2z12)).

Principal values on each side of these equations correspond. For these and further results see Miller (1955, pp. 42–43 and 77–79).

Confluent Hypergeometric Functions

10.39.5 Iν(z) =(12z)νe±zΓ(ν+1)M(ν+12,2ν+1,2z),
10.39.6 Kν(z) =π12(2z)νe-zU(ν+12,2ν+1,2z),
10.39.7 Iν(z)=(2z)-12M0,ν(2z)22νΓ(ν+1),
10.39.8 Kν(z)=(π2z)12W0,ν(2z).

For the functions M, U, M0,ν, and W0,ν see §§13.2(i) and 13.14(i).

Generalized Hypergeometric Functions and Hypergeometric Function

10.39.10 Iν(z)=(12z)νlimF(λ,μ;ν+1;z2/(4λμ)),

as λ and μ in , with z and ν fixed. For the functions F10 and F see (16.2.1) and §15.2(i).