Legendre polynomials 570 INDEX

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

Lumme and Rahola [53] considered cometaiy particles as stochastically shaped, i.e., particles whose shape can be described by a mean radius and the covariance function of the radius given as a series of Legendre polynomials. They made computations for a variety of particle shapes and size parameters (x = 16) using the refractive index m = 1.5 + i0.005. They found that the particles should have size parameters x > 1 to provide the negative polarization and low maximum polarization. Ensembles of particles with a power-law size distribution showed phase functions of intensity and polarization similar to the cometaiy ones. No information of the spectral characteristics was presented. 

The 0 part of the differential in equation 11.46 does have a known solution. The solution is a set of functions known as associated Legendre polynomials. (As with the Hermite polynomials, differential equations of the form in equation 11.46 had been previously studied, by the French mathematician Adrien Legendre, but for different reasons.) These polynomials, listed in Table 11.3, are functions of 0 only, but have two indices labeling the functions. One of the indices, an integer denoted , indicates the maximum power, or order, of 0 terms. (It also indicates the total order of the combination of cos 0 and sin 0 terms.) The second index, m, specifies which... 

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