Legendre’s functions

In studying the attraction and gravitational fields of the earth it is very useful to represent the potential in terms of Legendre s functions and this is mainly related to the fact that the earth is almost spherical and the position of observation points is... 

First, we introduce Legendre s functions, considering the power series of the function... 

In principle, in the same manner we can obtain the Legendre s functions of any order n. 

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. 

Let us demonstrate that with help of Legendre s functions we can find a solution of Laplace s equation. As is well known, Laplace s equation has the following form in the spherical system of coordinates ... 

The second equation of this set is a known equation (Chapter 1) and its solutions are Legendre s functions P (t]) and Now we demonstrate that the first... 

This function is a solution of Laplace s equation regardless of the values of constants, and our goal is to find such of them that the potential satisfies the boundary condition on the surface of the given ellipsoid of rotation and at infinity. In order to solve this problem we have to discuss some features of Legendre s functions. First of all, as was shown in Chapter 1, the Legendre s function of the first kind P (t]) has everywhere finite values and varies within the range... 

The Legendre s function of the second kind g (> ) has completely different behavior in particular, it tends to infinity when = 1. In accordance with Equation (2.121), this happens at points of the z-axis. Since the potential has everywhere a finite value the function Qn(ri) cannot describe the attraction field and has to be removed from Equation (2.132). This first simplification gives ... 

The latter can be also simplified, and with this purpose in mind consider the behavior of Legendre s functions with an imaginary argument. As follows from... 

The Legendre s function of the second kind has everywhere finite values, except = 0. 

There is also simple recursive expression which allows one to find this function of any order n, if we know the Legendre s functions of the first kind. Now we will demonstrate that the functions 2 (/g) tend to zero when the distance from the ellipsoid, (g = go), increases. Making use of the power series... 

Now we make use of the orthogonality of Legendre s functions. This means that from the equality... 

Earlier we solved the boundary value problem for the spheroid of rotation and found the potential of the gravitational field outside the masses provided that the outer surface is an equipotential surface. Bearing in mind that, we study the distribution of the normal part of the field on the earth s surface, where the position of points is often characterized by spherical coordinates, it is natural also to represent the potential of this field in terms of Legendre s functions. This task can be accomplished in two ways. The first one is based on a solution of the boundary value problem and its expansion into a series of Legendre s functions. We will use the second approach and proceed from the known formula, (Chapter 1) which in fact originated from Legendre s functions... 

Here the point p belongs to the spherical surface A of radius R. In order to find the upper limit on the left hand side of this equality, let us recall that T is the disturbing potential. In other words, it is caused by the irregular distribution of masses whose sum is equal to zero. This means that its expansion in power series with Legendre s functions does not contain a zero term. The next term is also equal to zero, because the origin coincides with the center of mass. Therefore, the series describing the function T starts from the term, which decreases as r. This means that the product r T O if oo and... 

Legendre s functions of the first and second kind with real argument Legendre s functions with imaginary argument radius of sphere radius vectors... 

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