Digital Library of Mathematical Functions
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NIST
10 Bessel FunctionsModified Bessel Functions

§10.34 Analytic Continuation

When m,

10.34.1 Iν(zmπ)=mνπIν(z),
10.34.2 Kν(zmπ)=-mνπKν(z)-πsin(mνπ)csc(νπ)Iν(z).
10.34.3 Iν(zmπ) =(/π)(±mνπKν(z±π)(m1)νπKν(z)),
10.34.4 Kν(zmπ) =csc(νπ)(±sin(mνπ)Kν(z±π)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(zmπ)=(-1)mnKn(z)+(-1)n(m-1)-1mπIn(z),
10.34.6 Kn(zmπ)=±(-1)n(m-1)mKn(z±π)(-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.