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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 Laplace’s equation, ∇ 2 V = 0 (1) in spherical coordinates ( r, θ, φ ): 1 r 2 ∂ ∂r r 2 ∂V ∂r ¶ + 1 r 2 sin θ ∂ ∂θ sin θ ∂V ∂θ ¶ + 1 r 2 sin 2 θ ∂ 2 V ∂φ 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 φ , and hence the Laplace’s equation reduces to: 1 r 2 ∂ ∂r r 2 ∂V ∂r ¶ + 1 r 2 sin θ ∂ ∂θ sin θ ∂V ∂θ ¶ = 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,θ ) = R ( r )Θ( θ ) , (4) where R ( r ) is a function of r alone and Θ( θ ) is a function of θ only. As we substitute this ansatz, partial derivatives become total derivatives: Θ r 2 d dr r 2 dR dr ¶ + R r 2 sin θ d dθ sin θ d Θ dθ ¶ = 0 . (5) 1 Multiplying this equation by r 2 and dividing by R Θ, we get and the equation can be written as two separate differential equations that are equal to the same constant,...
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This note was uploaded on 01/28/2010 for the course PHYS 351 taught by Professor Gangopashyaya during the Spring '09 term at Loyola Chicago.
- Spring '09