Spherical modified Bessel functions

The properties of the modified spherical Bessel functions (22) and Equation 33... 

This equation can be solved by Laplace transform techniques and Mt expressed as modified spherical Bessel functions [28]. However, because the boundary conditions on M are radically symmetric, only the / = 0 (i.e. S-wave) component is of interest. 

Again, the solutions of this equation involve the modified spherical Bessel functions and are... 

In Eq. (6) the modified spherical Bessel functions of third kind k,(z)10 are used. The total potential in the surrounding media can be written as follows... 

Equation (109) can readily be derived from Eq. (110). The functions zn and kn are modified spherical Bessel functions of order w, and P (x) are the generalized Legendre polynomials [91,97]. 

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 and yi are regular and irregular spherical Bessel functions,and ii and ki are modified spherical Bessel functions of the first and third kinds,respectively. Eq. (6.16) reduces to eq. (6.14) in the far asymptotic region. From the open-open sub-block of one can obtain the corresponding sub-block of the... 

The aji are real and closely related to the modified spherical Bessel functions 7/ + which are regular at the origin, and the 6/ are also real and closely related to the modified spherical Bessel functions which die exponentially at large distances. 

This equation is well known in applied mathematics, whose solutions involve modified spherical Bessel functions of the third kind. The solution for d = 1 differs from those for d>2 m the latter case, one distinguishes between two limits ... 

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