Laguerre integration

Although it is, in principle, possible to evaluate (6.2.11) analytically, it is numerically more advantageous to evaluate the integral in (6.2.11) with the help of a suitable Gauss-Laguerre integration formula (see, e.g., Stroud and Secrest (1966)). 

It is shown in Appendix 6 that the generalized Laguerre polynomials are eigenfunctions of the integral operator (3.26) with kernel (3.52). Let us search for the solution of (3.26) in the form of expansion over these eigenfunctions... 

G is then a generating function for these integrals, which occur as coefficients in its expansion in powers of u and and it can he evaluated with the use of the generating function for the associated Laguerre polynomials, given in equation (19). Thus we have... 

Hie growth integral term has been calculated through a numerical Oauss-Laguerre cpiadrature (JJ). All the moments appearing in equation 36 with order different from k = 1,M are evaluated... 

In order to obtain the orthogonality and normalization relations of the assoeiate Laguerre polynomials, we make use of the generating fimction (F. 10). We multiply together g(p, s j), g(p, t J), and the factor pj+ e-f and then integrate over p to give an integral that we abbreviate with the symbol /... 

The following integral from Gradshteyn and Ryzhik (1980, pp. 843-848) is useful in the evaluation of the Fourier transform of a Laguerre function as well as its convolution with a complex-valued Gaussian (Martin and Bowen, 1993) ... 

The use of a finite-basis expansion to represent the continuum is reminiscent of the use of quadratures to represent an integration. Heller, Reinhardt and Yamani (1973) showed that use of the Laguerre basis (5.56) is equivalent to a Gaussian-type quadrature rule. The underlying orthogonal polynomials were shown by Yamani and Reinhardt (1975) to be of the Pollaczek (1950) class. 

Three related approaches to the calculation of the recombination rate constant are described subsequently, the second two being approximations to the first. In each case the integral over E and the sum over J in the numerator of Eq. (3.3) are approximated by /V-point Laguerre and 2M-point extended Simpson s rule quadratures,30 respectively, yielding... 

In Appendix VII, it is shown that the normalization integral for the associated Laguerre function has the value... 

To this aim the integral representation of Laguerre polynomial is most useful it is based on complex integral representation of the solution of the Laguerre original differential equation ... 

In the same way as proceeded with the Laguerre s polynomial, the Hermite equation may be solved by the speeifie eomplex integral representation ... 

Therefore, in the light of the Laurent theorem of residues there can be inferred that the complex integral solution of the Hermite equation may produce the Hermite generating function (with the same recipe as was previously done for generating Laguerre function) ... 

Rewrite the appropriate parts of the UTILITY routines to incorporate the much more efficient Laguerre quadrature [64.H 67.H], rather than the strip integration which is now used to calculate the rate constant... 

---