Bessel functions computation

Bessel function 40, 99, 201, 264 binary approximation 7, 41 binary collisions adiabatic/non-adiabatic 4 angular momentum, computer simulations 40... 

After making these adjustments to allow for the fact that the analysis line cannot be located in the region of space where the centrifugal coupling in the body-fixed coordinates is negligible, and also for the fact that the analysis of Ref. 75 did not account for the long-range analytic form of the spherical Bessel functions, the space-fixed S matrix of Eq. (4.47) must be transformed back to the body-fixed axes and Eq. (4.46) must be used to compute the state-to-state differential cross sections [136,160]. 

Computation of Bessel Functions The Bessel functions Jn and Yn pose more computational problems than the logarithmic derivative. In BHCYL these functions are computed by an algorithm credited to Miller (British Association, 1952, p. xvii), further details of which are given by Stegun and Abramo-witz (1957) and by Goldstein and Thaler (1959) we outline this scheme in the following paragraph. 

Tests of BHCYL Computed values of J (x) and Yn(x) for various n and x were compared with values tabulated by Olver (1964). Computed logarithmic derivatives were also compared with values calculated from tabulated Bessel functions. In all instances there was agreement to as many decimal places as were given in the tables. This gives us some confidence that BHCYL does what it was designed to do. 

BESSEL FUNCTIONS J(N) COMPUTED BY DOWNWARD RECURRENCE BEGINNING AT N = NSTOP + NDELTA... 

Goldstein, M., and R. M. Thaler, 1959. Recurrence techniques for the calculation of Bessel functions, Math. Tables Other Aids Comput., 13, 102-108. 

Stegun, I. A., and M. Abramowitz, 1957. Generation of Bessel functions on high speed computers, Math. Tables Other Aids Comput., 11, 255-257. 

By computation or by using tables of Bessel functions, values of CjA can be found for a range of irf2) /k /D t values. Let... 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

TABLE 11.2 Bessel Functions K0(kx) and Kx(kx) for Computing Potentials of Long Cylinders3... 

Our interest is in computing the transition probability at resonance, a condition which is met when WA = WB + mco where m is an integer. First, we explicitly write the matrix element (V bWMV aW) using the wavefunctions of Eqs. (15.16) and (15.17) and the Bessel function relation of Eq. (15.15). It is given by... 

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). 

Some of the electric potential distributions in a two-dimensional set up can be expressed in infinite series terms, when other mathematical algorithms do not work. Special series terms of Bessel functions and Legendre polynomials are easily evaluated nowadays with the help of computer... 

This defines a Bessel function of the second kind (sometimes called a Neumann function, written as N ) The computational details are horrible, but... 

The modified Bessel functions can be computed from the general result for any p... 

We need the differential or integral properties for Bessel functions to compute the net rate of heat transfer. We discuss these properties in the next section, and then use them to complete the above example. 

For the Lekner method, upper error boimds can be given easily since the Bessel functions drop essentially exponentially fast, allowing for a simple approximation of the sum by an integral. These error estimates are much less sharp than the error estimates for the Ewald-type methods, but here the error bound only enters logarithmically into the computation time, so that excessive accuracy has only a small impact on the overall performance. 

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