v] is the modified Bessel function
of second kind of order v.
n] = nth-order modified Bessel function
of first kind
The solution of equation (1) has been presented in  based on modified Bessel function
O] is a modified Bessel function
of first kind and zero order and [K.
As a general example, modified expansions for confluent hypergeometric functions are considered in Section 3, and as particular cases expansions for the incomplete gamma function [GAMMA](a, z) and the modified Bessel function
v] (x) is the modified Bessel function
of the second kind (also called MacDonald function).
This edition, revised from the 2009 seventh, includes eight new projects, updated exercise sets, additional examples and figures, a simplified account of linear first-order differential equations, new sections on Green's function and the review of power series, and several boundary-value problems involving modified Bessel functions
10), which involves the modified Bessel functions
This edition has a new section on Green's functions for linear ordinary differential equations; expanded information on modified Bessel functions
and problems in cylindrical coordinates; rewritten sections on dot and cross products and independence of path; new problems; and nine new projects.
Since the modified Bessel functions
are exponentially suppressed at infinity, the only contribution comes from the lower limit
N x x > 0 K and 0 I are the modified Bessel functions
of zero order with a complex argument.
Appendixes cover topics that are not standard parts of an average science of engineering undergraduate curriculum: Green's functions, real spherical harmonics, spherical modified Bessel functions
, plane groups and invariant functions, and elementary elasticity theory.