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10 Bessel FunctionsModified Bessel Functions

§10.27 Connection Formulas

Other solutions of (10.25.1) are I-ν(z) and K-ν(z).

10.27.1 I-n(z)=In(z),
10.27.2 I-ν(z)=Iν(z)+(2/π)sin(νπ)Kν(z),
10.27.3 K-ν(z)=Kν(z).
10.27.4 Kν(z)=12πI-ν(z)-Iν(z)sin(νπ).

When ν is an integer limiting values are taken:

10.27.5 Kn(z)=(-1)n-12(Iν(z)ν|ν=n+Iν(z)ν|ν=-n),
n=0,±1,±2,.

In terms of the solutions of (10.2.1),

10.27.6 Iν(z)=eνπi/2Jν(ze±πi/2),
-π±phz12π,
10.27.7 Iν(z)=12eνπi/2(Hν(1)(ze±πi/2)+Hν(2)(ze±πi/2)),
-π±phz12π.
10.27.8 Kν(z)={12πieνπi/2Hν(1)(zeπi/2),-πphz12π,-12πie-νπi/2Hν(2)(ze-πi/2),-12πphzπ.
10.27.9 πiJν(z)=e-νπi/2Kν(ze-πi/2)-eνπi/2Kν(zeπi/2),
|phz|12π.
10.27.10 -πYν(z)=e-νπi/2Kν(ze-πi/2)+eνπi/2Kν(zeπi/2),
|phz|12π.
10.27.11 Yν(z)=e±(ν+1)πi/2Iν(zeπi/2)-(2/π)eνπi/2Kν(zeπi/2),
-12π±phzπ.

See also §10.34.

Many properties of modified Bessel functions follow immediately from those of ordinary Bessel functions by application of (10.27.6)–(10.27.8).