Analog Bessel function

This is the mass transfer analog of the L v6que problem in heat transfer (S9, p. 81), and the solution is expressible in terms of Bessel functions (C2, p. 175) ... 

An imaginary argument is also possible for Bessel functions. When this occurs, they become the modified Bessel functions I and K. This substitution changes them from oscillatory to monotonic, as in the analogous case of the trigonometric functions. The modified Bessel function of the first is defined as... 

This series arises naturally, when expressing the Coulomb potential of a charge separated by a distance s from the origin in terms of spherical coordinates. The positive powers result when r < s, while for r > s the potential is described by the negative powers. Similarly the solutions of the linearized Poisson-Boltzmann equation are generated by the analogous expansion of the shielded Coulomb potential exp[fix]/r of a non-centered point charge. Now the expansion for r > s involves the modified spherical Bessel-functions fo (x), while lor r < s the functions are the same as for the unshielded Coulomb potential,... 

The two kinds of Bessel functions, thus, have the asymptotic dependence of slowly damped cosines and sines. In analogy with Euler s formula = cos X i sinx, we define Hankel functions of the first and second kinds ... 

In NQR, however, the sample usually consists of a polycrystalline powder. Then a convolution for all orientations must be made. The result O is that the sin i ty,) function in the expression for the expectation value of the magnetization becomes a Bessel function, Ji( /2wi t ). This function has its first maximum, analogous to a "90 pulse", at ty, = 0.66 and not at... 

At fixed 5, the terms in (5.180) represent the modes of the f-shape of the potential. The Bessel function oscillates and thus a s are analogous to frequencies in the Fourier series expansion. Equation (5.180) shows that each mode exponentially decays with the distance 5 the characteristic damping length is / = l/a , or, in the dimensional form... 

As we see, special Bessel functions are included in the equations. Despite an outward difficult look of those equations they are completely usable to calculate kinetic curves of substances B and C. Bessel functions values for given parameters values could be computed with the help of mathematic packages. Let us pint, that Maple s built-in functions Bessell(0,x), Bessell(l,x), BesselK (0,x), BesselK(l,x) have their Mathcad-analogs in the form of built-in functions I0(x), Il(x), K0(x), Kl(x) correspondingly. 

In the framework of our formalism, the description of this resonant interaction is essentially analogous to that of inelastic diffraction ( washboard -phonon resonance, see brief discussion in the previous section). The trajectories of translational and rotational motion consist of two elastic branches as well, and the result may be expressed through the Bessel functions of the corresponding classical action increments. 

The first of these, which connects the logarithmic derivatives of the spherical Bessel and Neumann functions, is a direct consequence of the Wronskian relation nj - jn = S [5.4], while the latter may be derived quite generally in analogy to (4.17), to which it reduces when the two boundary conditions are - i - 1 and . In addition, we renormalise the structure constants... 

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