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10 Bessel FunctionsModified Bessel Functions

§10.34 Analytic Continuation

When m,

10.34.1 Iν(zemπi)=emνπiIν(z),
10.34.2 Kν(zemπi)=e-mνπiKν(z)-πisin(mνπ)csc(νπ)Iν(z).
10.34.3 Iν(zemπi) =(i/π)(±emνπiKν(ze±πi)e(m1)νπiKν(z)),
10.34.4 Kν(zemπi) =csc(νπ)(±sin(mνπ)Kν(ze±πi)sin((m1)νπ)Kν(z)).

If ν=n(), then limiting values are taken in (10.34.2) and (10.34.4):

10.34.5 Kn(zemπi)=(-1)mnKn(z)+(-1)n(m-1)-1mπiIn(z),
10.34.6 Kn(zemπi)=±(-1)n(m-1)mKn(ze±πi)(-1)nm(m1)Kn(z).

For real ν,

10.34.7 Iν(z¯) =Iν(z)¯,
Kν(z¯) =Kν(z)¯.

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