Legendre s equation

Letting n = n n + 1) and choosing the minus sign at the right hand side of the second equality we obtain Legendre s equation one of its solutions is the function... 

Legendre s equation has a second solution, but it is not considered here. As follows from Equation (1.164) for each n we have a partial solution, which can be represented in the form ... 

The relationship between (23) and Legendre s equation can be demonstrated in two steps. First, substitute... 

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

For m an integer the P/"(x) are polynomials. The polynomials P,°(x) are identical with the Legendre polynomials. The first few associated Legendre polynomials for x = cos 9 (a common form of Legendre s equation) are ... 

With these definitions we may write the solution of Legendre s equation (17.1) as... 

Since Legendre s equation has a regular singular point at ft = 1 we nmy on the basis of equation (3.8) take the... 

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

In the method of orthogonal collocation to be described later, Villadsen and Michelsen (1978) define Legendre s equation as... 

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

Therefore, one can obtain two linearly independent solutiwis of Legendre s equation. Say for = 1 and ai = 0... 

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