Laguerre

The functions P " are associated Legendre polynomials of order m and degree I, and are associated Laguerre polynomials of degree (v — l)/2 in... 

These functions are universally known as Slater type orbitals (STOs) and are just the leading term in the appropriate Laguerre polynomials. The first three Slater functions are as follows ... 

Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

Methods such as Graeffe s root-squaring method, Muller s method, Laguerre s method, and others exist for finding all roots of polynomials with real coefficients. [12 and others]... 

Hol0ien, E., Phys. Rev. 104, 1301, Radial configurational interaction in He and similar atomic systems." An orthonormal set of associated Laguerre functions is used. 

Stewart, A. L., Proc. Phys. Soc. A70, 756, "Wave functions for He and similar atomic systems. Perturbation treatment of second order, based on complete set of Laguerre functions. 

Hol0IEN, E., Proc. Phys. Soc. A71, 357, "The 2s2 XS state solution of the non-relativistic Schrodinger equation for He and H. " Associated Laguerre functions. 

The Sonine polynomials are related to the associated Laguerre polynomials (see Margenau and Murphy, op. eft., p. 128) by... 

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... 

As the isotropic Raman and CARS spectra may be expressed via ao(co) by virtue of the Laguerre polynomials orthogonality, we have... 

Now it is possible to employ the eigenfunctions given in [121], which are proportional to the generalized Laguerre polynomials. The corresponding eigenvalues are equidistant ... 

Undetermined coefficients 4 can be easily found using the initial condition (6.3) and orthogonality of the Laguerre polynomials [37]... 

In a general case parameters re, XdP and y must be determined by self-consistent two-parameter fitting. Owing to the property of orthogonality of Laguerre polynomials, one has for the spectral band shapes... 

Laguerre polynomials 3-D. /-space 264 eigenfunctions 119, 262-3 Raman and CARS spectra 120 Landau-Teller formula 159 Langevin model equation 32... 

Let ) ( ) represents the (21 + l)th derivative of the (n + l)th Laguerre polynomial (20) and P7 (cos ) is Ferrers associated Legendre function of the first kind, of degree l and order m. Yim Zm thus constitutes a tesseral harmonic (21). The p s are in this form orthogonal and normalized, so that they fulfill the conditions... 

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... 

The following relations involving the A s can be easily derived from the properties of the associated Laguerre polynomials ... 

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... 

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