Legendre functions special values

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

In Eq. (5.6), we have connected in a special way the normalization parameter A with the Legendre multiplier and thus, we have specified the normalization of the function ip (compare with24 ). The value of <3E/<3n plays the role of the self-consistent field in Eq. (5.6). 

The boundary conditions are in general of the mixed type involving a combination of the function value and derivative at the two boundaries taken here to occur tx = a andx = b. Special cases of this equation lead to many classical functions such as Bessel functions, Legendre polynomials, Hemite polynomials, Laguerre polynomials and Chebyshev polynomials. In addition the Schrodinger time independent wave equation is a form of the Sturm-Liouville problem. 

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