Digital Library of Mathematical Functions
About the Project
NIST
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)=νπ/2Jν(z±π/2),
-π±phz12π,
10.27.7 Iν(z)=12νπ/2(Hν(1)(z±π/2)+Hν(2)(z±π/2)),
-π±phz12π.
10.27.8 Kν(z)={12πνπ/2Hν(1)(zπ/2),-πphz12π,-12π-νπ/2Hν(2)(z-π/2),-12πphzπ.
10.27.9 πJν(z)=-νπ/2Kν(z-π/2)-νπ/2Kν(zπ/2),
|phz|12π.
10.27.10 -πYν(z)=-νπ/2Kν(z-π/2)+νπ/2Kν(zπ/2),
|phz|12π.
10.27.11 Yν(z)=±(ν+1)π/2Iν(zπ/2)-(2/π)νπ/2Kν(zπ/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).