(2) For b = 1, c = -1, we obtain the
modified Bessel functions of first kind of order v whose series form is given as
[J.sub.m] and [Y.sub.m] are the Bessel functions of the first and second kinds and [I.sub.m] and [K.sub.m] are
modified Bessel functions of the first and second kinds, respectively.
where the argument of the
modified Bessel functions [I.sub.v](*) and [K.sub.v] (*) is j[k.sub.1][rho]/2, and
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. A shorter version covers only ordinary differential equations, for a one-semester or one-quarter course.
where G(r, r') is the Green function of (3.10), which involves the
modified Bessel functions [I.sub.1]([square root of [[beta].sub.n]] r) and [K.sub.1]([square root of [[beta].sub.n]] r).
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.
This relation is similar to the relation between Bessel and
modified Bessel functions.
where [mu] = ([lambda.sub.uo] [lambda.sub.ou][t.sub.u][t.sub.o])[sup.1/2], and [I.sub.o] (.) and .[I.sub.1] (.) are
modified Bessel functions of the first kind of order zero and one, respectively.
To obtain a solution of Equation 1 involves calculating Bessel functions and
modified Bessel functions. Methods that use numerical techniques to solve these functions on calculators or on desktop computers have involved the use of an algorithm that detracts from the accuracy.
where the argument of the
modified Bessel functions has been omitted for notational simplicity.
