Legendre associated

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

The functions are the associated Legendre polynomials of which a few are given in Table 1.1. They are independent of Z, the nuclear charge number, and therefore are the same for all one-electron atoms. 

M. A. Clarke and B. L. Legendre, Proceedings of the South African Sugar Technologists Association, 1996, in press. 

More simply, the chemical potentials appear as Legendre multipliers when G is minimised, associated with the constraints on total numbers of atoms, eg. (23). The results are ... 

The spherical harmonics are defined in terms of the associated Legendre polynomials, of variable cos 6, and exponential functions in... 

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... 

Relationship of spherical harmonics to associated Legendre polynomials... 

Equation (E.13) relates the associated Legendre polynomial Pfip) to the (/ -I- w)th-order derivative in equation (5.58)... 

The associated Legendre polynomials P l ip) are defined in terms of the Legendre polynomials Piip) by... 

The generating functions g " p, s) for the associated Legendre polynomials may be found from equation (E.l) by letting... 

Equation (E.17) is the associated Legendre differential equation. Orthogonality... 

An alternative definition, but equally useful, of the associated Legendre polynomials is of die form... 

When the associated Legendre polynomials are normalized they are written in the form... 

The explicit form of the normalized associated Legendre polynomials is given by... 

The associated Legendre polynomials can be defined by the generating function... 

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