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

§10.4 Connection Formulas

Other solutions of (10.2.1) include J-ν(z), Y-ν(z), H-ν(1)(z), and H-ν(2)(z).

10.4.1 J-n(z) =(-1)nJn(z),
Y-n(z) =(-1)nYn(z),
10.4.2 H-n(1)(z) =(-1)nHn(1)(z),
H-n(2)(z) =(-1)nHn(2)(z).
10.4.3 Hν(1)(z) =Jν(z)+Yν(z),
Hν(2)(z) =Jν(z)-Yν(z),
10.4.4 Jν(z) =12(Hν(1)(z)+Hν(2)(z)),
Yν(z) =12(Hν(1)(z)-Hν(2)(z)).
10.4.5 Jν(z)=csc(νπ)(Y-ν(z)-Yν(z)cos(νπ)).
10.4.6 H-ν(1)(z) =νπHν(1)(z),
H-ν(2)(z) =-νπHν(2)(z).
10.4.7 Hν(1)(z)=csc(νπ)(-νπJν(z)-J-ν(z))=csc(νπ)(Y-ν(z)--νπYν(z)),
10.4.8 Hν(2)(z)=csc(νπ)(J-ν(z)-νπJν(z))=csc(νπ)(Y-ν(z)-νπYν(z)).

In (10.4.5), (10.4.7), and (10.4.8) limiting values are taken when ν=n; compare (10.2.3) and (10.2.4).

See also §10.11.