The Associated Legendre Functions

The Associated Legendre Functions.—We define the associated Legendre functions of degree l and order mj (with values l — 0, 1, 2, and m = 0, 1, 2, , l) in terms of the Legendre polynomials by means of the equation 

This result enables us to identify1 the functions of Section 18c (except for constant factors) with the associated Legendre functions, inasmuch as Equation 19-9 is identical with Equation 18-19, except that P(z) is replaced by P[m (z) and j8 is replaced by 1(1 + 1), which was found in Section 18c to represent the characteristic values of fi. Hence the wave functions in d corresponding to given values of the azimuthal quantum number l and the magnetic quantum number m are the associated Legendre functions Pjml(z). 

The associated Legendre functions are most easily tabulated by the use of the recursion formula 19-2 and the definition 19-7, together with the value P (z) = 1 as the starting point. A detailed discussion of the functions is given in Section 21. 

For some purposes the generating function for the associated Legendre functions is useful. It is found from that for the Legendre polynomials to be 

1 The identification is completed by the fact that both functions are formed from polynomials of the same degree. 

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Substitution into the Rodriques formula gives the associated Legendre functions in the form... 

The theta integration involves the associated Legendre functions. Using4... 

In these coordinates Laplace s equation separates such functions of the form P ( )P. (7l)(g )m(J satisfy it where are the associated Legendre functions. Thus for this interior problem (i.e. inside the spheroid, not outside it) general solution of the linear problem is of the form... 

P 1 denotes the associated Legendre function of degree / and order m. Nt m is a normalization constant. 

The sum over wave numbers is written as an integral over wave number spaee, with a density of states in wave number space of 2/(2rc)- = Naii/ 6n ). The exponent k d is simply kd cos 0, and the spherical harmonics can be written out in terms of the associated Legendre functions, PtJ (cos (Schiff, 1968, p. 80),... 

A linear combination of the bound and continuum atomic wavefiinction was used to approximate the continua (10). Wavefunctions in the spherical potential were separated into the spherical harmonics and the radial wavefunctions. The spherical harmonics are expressed in terms of the associated Legendre functions. The differential equation for the radial wavefunction R at position r is... 

Pf z) are the associated Legendre functions of order l (m < l) [51], r = I/(2kBT), and the angular brackets ( ) denote ensemble averages over the distribution function W, namely,... 

Substituting Eq. (267) into Eq. (265), taking the inner product, and utilizing the orthogonal properties and known recurrence relations [51] for the associated Legendre functions Pf cosi ) and the Hermite polynomials H (z) then yields the infinite hierarchy of differential recurrence relations for the clnm(t) governing the orientational relaxation of the system, namely,... 

One may also readily derive differential-recurrence equations for the statistical moments involving the associated Legendre functions of order 2(1 = 2) pertaining to the dynamic Kerr effect, namely, b (t) [so that = (P2(005 d)) )]. 

Use of the orthogonality properties of the associated Legendre functions, which are... 

For this purpose, we note that the polynomials Qn(j]) can be expressed in terms of functions T (cos 6) that are known as the associated Legendre functions ... 

The visit to the north pole does not include any surprises. It involves the limit of infinitely large values of A., for which the associated Legendre functions become Bessel functions [16,22] of order m and argument AV2(1 — cos0). Correspondingly, its zeros become... 

The functions of 0 for m =0 are called Legendre functions. The associated Legendre functions include the Legendre functions and additional... 

It is to be noted that the order m is restricted to positive values (and zero) we are using the rather clumsy symbol m to represent the order of the associated Legendre function so that we may later identify m with the magnetic quantum number previously... 

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