Legendre functions polynomials

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... 

This is one of the main reasons why these functions play a very important role in solving boundary value problems. Also, between Legendre s polynomials of different order, there is a simple recursive relationship ... 

Let us notice that due to orthogonality of Legendre s polynomials many functions can be represented as a series, which is similar to Equation (1.162), and this fact is widely used in mathematical physics. Now, we will derive the differential equation, one of the solutions of which are Legendre s functions. 

The variation of Pn(ft) with ft for a few values of n is sliown in Fig. 5. Since, in most physical problems, the Legendre polynomial involved is usually / tl(cos 0) we have shown in Fig. fi the variation of this function with 0. Numerical values may be obtained from Tables of Associated Legendre Functions (Columbia University Press, 194-5),... 

Recurrence Relations for the Function Recurrence relations for the Legendre function of the second kind can be derived from Neumann s formula (18.1) anti the corresponding recurrence relations for the Legendre polynomials Pn(fi). From the recurrence relation (It.2) and Neumann s formula we have... 

The projection of T,p on each of the radial unit vectors can be evaluated in terms of the basic angular functions which make up the vector spherical harmonics.(27) Although these functions are associated Legendre polynomials for an arbitrarily oriented donor dipole, for the case of full azimuthal symmetry shown in Figure 8.19 the angular functions are ordinary Legendre functions, P (i.e., w = 0). Under these circumstances,... 

Physicists and chemists know the Legendre functions well. One very useful explicit expression for these functions is given in terms of derivatives of a polynomials. 

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,... 

Our experience with the hydrogen atom confined by a circular cone [39] involves going from Legendre polynomials P cos9) to Legendre functions... 

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 ... 

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

For w < 0 the phase factor (—1) should be omitted. The term in square brackets is a normalizing function, and pj (cos 6) represents some member of the series of associated Legendre functions. When m = 0, these become the ordinary Legendre polynomials. The first few ordinary Legendre polynomials are... 

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