Functions and a Solution of Laplaces Equation

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 

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

It is essential that in both cases this parameter is less than unity. Now we expand the function 

Differentiating the funetion (p with respeet to p and substituting p — 0, we obtain 

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

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