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10 Bessel FunctionsBessel and Hankel Functions

§10.4 Connection Formulas

Other solutions of (10.2.1) include Jν(z), Yν(z), Hν(1)(z), and Hν(2)(z).

10.4.1 Jn(z) =(1)nJn(z),
Yn(z) =(1)nYn(z),
10.4.2 Hn(1)(z) =(1)nHn(1)(z),
Hn(2)(z) =(1)nHn(2)(z).
10.4.3 Hν(1)(z) =Jν(z)+iYν(z),
Hν(2)(z) =Jν(z)iYν(z),
10.4.4 Jν(z) =12(Hν(1)(z)+Hν(2)(z)),
Yν(z) =12i(Hν(1)(z)Hν(2)(z)).
10.4.5 Jν(z)=csc(νπ)(Yν(z)Yν(z)cos(νπ)).
10.4.6 Hν(1)(z) =eνπiHν(1)(z),
Hν(2)(z) =eνπiHν(2)(z).
10.4.7 Hν(1)(z)=icsc(νπ)(eνπiJν(z)Jν(z))=csc(νπ)(Yν(z)eνπiYν(z)),
10.4.8 Hν(2)(z)=icsc(νπ)(Jν(z)eνπiJν(z))=csc(νπ)(Yν(z)eνπiYν(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.