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Class_15new - PHYS809 Class 15 Notes Green function in...

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1 PHYS809Class15Notes Green function in spherical polar coordinates To illustrate construction of a Green function in spherical polar coordinates consider the Dirichlet problem in a region bounded by two concentric sphere of radii a and b with a < b . The Green function is the solution of ( ) ( ) ( ) ( ) 2 2 4 , , , , , cos cos , G r r r r r π θ φ θ φ δ δ θ θ δ φ φ = - - - - (15.1) with G = 0 on r = a and r = b . Since the completeness relation for the spherical harmonics is ( ) ( ) ( ) ( ) * 0 , , cos cos , l lm lm l m l Y Y θ φ θ φ δ φ φ δ θ θ = =- = - - ∑ ∑ (15.2) we look for a solution of form ( ) ( ) ( ) ( ) * 0 , , , , , , , , . l lm lm lm l m l G r r g r r Y Y θ φ θ φ θ φ θ φ = =- = ∑ ∑ (15.3) Remembering that ( ) , l lm r Y θ φ is a solution of Laplace’s equation, substitution of (15.3) into equation (15.1) gives ( ) ( ) ( ) ( ) ( ) ( ) 2 * * 2 2 2 0 0 1 1 4 , , , , . l l lm lm lm lm lm lm l m l l m l l l r g g Y Y r r Y Y r r r r r π θ φ θ φ δ θ φ θ φ = =- = =- + - = - - ∑ ∑ ∑ ∑ (15.4) Since the spherical harmonics are linearly independent, we must have ( ) ( ) 2 2 2 2 1 1 4 . lm lm l l r g g r r r r r r r π δ + - = - - (15.5) The homogeneous equation has solution of form ( ) 1 , , l l lm mn mn g r r A r B r - - ′ = + (15.6) where the coefficients will depend on . r Because of the boundary conditions at r = a and r = b , the solution is of form
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2 ( ) 1 1 , , , , . l l lm l l r a A a r r a r g r r r b A r r b b r + < + > - < ′ = - < (15.7)
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