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10 Bessel FunctionsBessel and Hankel Functions

§10.11 Analytic Continuation

When m,

10.11.1 Jν(zemπi)=emνπiJν(z),
10.11.2 Yν(zemπi)=e-mνπiYν(z)+2isin(mνπ)cot(νπ)Jν(z).
10.11.3 sin(νπ)Hν(1)(zemπi)=-sin((m-1)νπ)Hν(1)(z)-e-νπisin(mνπ)Hν(2)(z),
10.11.4 sin(νπ)Hν(2)(zemπi)=eνπisin(mνπ)Hν(1)(z)+sin((m+1)νπ)Hν(2)(z).
10.11.5 Hν(1)(zeπi) =-e-νπiHν(2)(z),
Hν(2)(ze-πi) =-eνπiHν(1)(z).

If ν=n (), then limiting values are taken in (10.11.2)–(10.11.4):

10.11.6 Yn(zemπi)=(-1)mn(Yn(z)+2imJn(z)),
10.11.7 Hn(1)(zemπi)=(-1)mn-1((m-1)Hn(1)(z)+mHn(2)(z)),
10.11.8 Hn(2)(zemπi)=(-1)mn(mHn(1)(z)+(m+1)Hn(2)(z)).

For real ν,

10.11.9 Jν(z¯) =Jν(z)¯, Yν(z¯) =Yν(z)¯,
Hν(1)(z¯) =Hν(2)(z)¯, Hν(2)(z¯) =Hν(1)(z)¯.

For complex ν replace ν by ν¯ on the right-hand sides.