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Unformatted text preview: Legendre Polynomials and their properties Original by Gregory E. Amenta, Modified by A. Gangopadhyaya March 5, 2006 Let us consider Laplaces equation, 2 V = 0 (1) in spherical coordinates ( r, , ): 1 r 2 r r 2 V r + 1 r 2 sin sin V + 1 r 2 sin 2 2 V 2 = 0 . (2) At this point, let us only consider cases where there is an azimuthal symme- try; i.e., there is an axis about which the system is invariant under rotation. Let us choose this axis of symmetry to be the z-axis; this makes the poten- tial to be independent of the coordinate , and hence the Laplaces equation reduces to: 1 r 2 r r 2 V r + 1 r 2 sin sin V = 0 . (3) To solve this partial differential equation, we would like to convert it into a set of ordinary differential equations by trying the ansatz: V ( r, ) = R ( r )( ) , (4) where R ( r ) is a function of r alone and ( ) is a function of only. As we substitute this ansatz, partial derivatives become total derivatives: r 2 d dr r 2 dR dr + R r 2 sin d d sin d d = 0 . (5) 1 Multiplying this equation by r 2 and dividing by R , we get and the equation can be written as two separate differential equations that are equal to the same constant,...
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- Spring '09