Bessel function of the second kind

In this equation, Jv is the Bessel function of the first kind, and Kv is the modified Bessel function of the second kind, U = aikfnf—fi1)112, W = a(f21—konf)if2,... 

The function F0(x), so defined is known as Weber s Bessel function of the second kind of zero order. 

The functions PJT(cos 9) are associated Legendre functions of the first kind of degree n and order m, and z (kr) denotes any of four spherical Bessel functions. The choice of the spherical Bessel function depends on the domain of interest, that is, on whether we are looking for the solution inside the sphere (r < a) or outside the sphere (r > a). For the internal field we choose z (kr) = j (kr), where j (kr) is the spherical Bessel function of the first kind of order n. The solution for the external field can be written in terms of spherical Bessel functions j kr) and y kr), where the latter is the spherical Bessel function of the second kind, but it is more convenient to introduce the spherical Hankel function /i / (kr) to determine tj/ for the outer field. 

K2(z) is the second-order, modified Bessel function of the second kind. For h — 0, this expression for a reduces to the classical result [232],... 

In practice, the important fact is that there are two independent solutions, J and Y, and Y is infinite at the origin. These Y s are called Bessel functions of the second kind. Some of these functions are shown in Figure A.2. 

With proper adjustments due to the factor i, these functions follow recurrence relations similar to those for J Io(0) = 1, In (0) = 0, and for n >0 the modified Bessel function is monotonically increasing. The second solution, K, does not follow the same recurrence relations as I. Macdonald s definition of the modified Bessel function of the second kind is... 

Similar expressions apply to the other types of pairs, and all of them contain the zeroth-order modified Bessel function of the second kind K0(xr), or, simply, the Bessel K0 function. In Figure 1 we show a graph of the smoothly decreasing nonoscillatory Bessel K0 function, which can subsequently be contrasted with the pair potentials w(r) that emerge only after the three free energy terms (polyion self-energy, counterion transfer, and direct pair interaction) are added and minimized, a procedure that involves determination of the functions 0(r) and Q(r). 

FIG. 1 The zeroth-order modified Bessel function of the second kind, K0(xr). 

Verify that the potential distribution about the cylinder takes the form (f) = AKo(r/Ajj), where r is the radial cylindrical coordinate and Kq is the modified Bessel function of the second kind of order zero. Evaluate the constant A. 

The modified zeroth-order Bessel function of the second kind approaches infinity in the limit of a zero argument lim, o( o) oo, which corresponds to the symmetry axis of the cylinder. Hence, one sets the integration constant C2 to zero and the boundary condition on the external surface of the catalyst is used to calculate Ci = l//o(A). The final solution for the basic information is given by equation (17-14). 

The first boundary condition is equivalent to a finite value of I a at the symmetry point in spherical coordinates. This condition was invoked in Section 17.2 along the symmetry axis of long cylindrical catalysts to eliminate the modified zeroth-order Bessel function of the second kind, = 0), from the general solution given by equation (17-22). When the symmetry condition at the center of a spherical pellet is used to evaluate the integration constants, one finds that B = 0 in equation (17-28) because ... 

Bessel functions of the second kind show the following Umiting behavior ... 

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