Legendre’s differential equation

The third and last terms on the left-hand side may be eliminated by means of equation (E.5) to give Legendre s differential equation... 

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 next step is to differentiate Legendre s equation n times. Using the Leibniz [6] theorem one gets... 

If we substitute from (18.8 into Legendre s equation (171. ) wc find that II7n 1 satisfies the differential equation... 

Legendre s Associated Functions. We saw in example 1 of Chapter 1 that the solution of Laplace s equation in spherical polar coordinates reduces to the solution of the ordinary differential equation,... 

In quantum mechanics and other branches of mathematical physics, we repeatedly encounter what are called special functions. These are often solutions of second-order differential equations with variable coefficients. The most famous examples are Bessel functions, which we wiU not need in this book. Our first encounter with special functions are the Hermite polynomials, contained in solutions of the Schrodinger equation. In subsequent chapters we will introduce Legendre and Laguerre functions. Sometime in 2004, theU.S. National Institute of Standards and Tec hnology (NIST) will publish an online Digital Library of Mathematical Functions, http / /dlmf. nist. gov, including graphics and cross-references. 

Three important examples of second-order Unear differential equatirms that frequently occur in chemical engineering are Legendre s equation [3,4] of order a... 

Spherical harmonics Y 9,(l>) are the angular contribution to the solution of Laplace s equation (or Helmholtz differential equation) in spherical coordinates (i.e. Eqs. (C.9) and (C.IO)). They are hence the product of the associated Legendre polynomial of cos0 and the general sine of the azimuth (/> ... 

The independent variables in a differential expression can be changed by a Legendre transformation. For example, to reexpress the fundamental equation in terms of S and p, rather than S and V, we define the enthalpy H=U+pV. This satisfies the differential relation... 

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