Ricatti-Bessel function

The generalisation of u o r) to arbitrary positive integers tf is given by the regular and irregular Ricatti—Bessel functions U( p) and V( p) respectively. They satisfy (4.12) with... 

For r > ro the solution is given by the appropriate linear combination of the regular and irregular Ricatti—Bessel functions of imaginary argument. Putting... 

For continuum channels the radial orbitals in (7.140) are not bounded and the integral is divergent. The choice of radial functions in (7.136) must be based on intuition obtained from a study of ionisation, which is treated in chapter 10. A necessary condition for a reasonable distorted wave in the distorted-wave Bom approximation for ionisation is that it should be orthogonal to the initial state in the ionisation amplitude. For computational simplicity we set Fop,7l equal to zero and orthogonalise the resulting Ricatti—Bessel functions to all the states of P space using (5.83). 

Here we have converted the Ricatti—Bessel functions of (7.142) to cylindrical Bessel functions in order to make use of analytic identities (Luke, 1973). 

Here a and b are series of Ricatti-Bessel functions [4]. The measured attenuation of light by a colloid (and indeed by molecular solutions) is always due to both scattering and absorption. These effects are additive so... 

Let us now introduce p via /3(/3 + 1) = a. Weakly attractive potentials correspond to -1/2 < /I < 0 and repulsive potentials to > 0. In both cases, the solution of the Schrodinger equation at large r can be written as a linear combination of the Ricatti-Bessel functions /(kr) and h/s(kr), and the order of the corresponding cylinder functions v = p + 1/2 remains positive. Using analytic properties of the Bessel functions as kr 0, to lowest order in k,... 

The computation of light scattering frequently employs the Ricatti-Bessel functions, which are related to the spherical Bessel functions ... 

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