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10 Bessel FunctionsModified Bessel Functions

§10.25 Definitions

Contents

§10.25(i) Modified Bessel’s Equation

10.25.1 z2d2wdz2+zdwdz-(z2+ν2)w=0.

This equation is obtained from Bessel’s equation (10.2.1) on replacing z by ±iz, and it has the same kinds of singularities. Its solutions are called modified Bessel functions or Bessel functions of imaginary argument.

§10.25(ii) Standard Solutions

10.25.2 Iν(z)=(12z)νk=0(14z2)kk!Γ(ν+k+1).

This solution has properties analogous to those of Jν(z), defined in §10.2(ii). In particular, the principal branch of Iν(z) is defined in a similar way: it corresponds to the principal value of (12z)ν, is analytic in \(-,0], and two-valued and discontinuous on the cut phz=±π.

The defining property of the second standard solution Kν(z) of (10.25.1) is

10.25.3 Kν(z)π/(2z)e-z,

as z in |phz|32π-δ (<32π). It has a branch point at z=0 for all ν. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in \(-,0], and two-valued and discontinuous on the cut phz=±π.

Both Iν(z) and Kν(z) are real when ν is real and phz=0.

For fixed z (0) each branch of Iν(z) and Kν(z) is entire in ν.

Branch Conventions

Except where indicated otherwise it is assumed throughout the DLMF that the symbols Iν(z) and Kν(z) denote the principal values of these functions.

Symbol 𝒵ν(z)

Corresponding to the symbol 𝒞ν introduced in §10.2(ii), we sometimes use 𝒵ν(z) to denote Iν(z), eνπiKν(z), or any nontrivial linear combination of these functions, the coefficients in which are independent of z and ν.

§10.25(iii) Numerically Satisfactory Pairs of Solutions

Table 10.25.1 lists numerically satisfactory pairs of solutions (§2.7(iv)) of (10.25.1). It is assumed that ν0. When ν<0, Iν(z) is replaced by I-ν(z).

Table 10.25.1: Numerically satisfactory pairs of solutions of the modified Bessel’s equation.
Pair Region
Iν(z),Kν(z) |phz|12π
Iν(z),Kν(zeπi) 12π±phz32π