Chapter5.p44.Greens.function

Chapter5.p44.Greens.function - Boundary Value Problems in...

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Boundary Value Problems in Spherical Coordinates Spherical harmonics Laplace’s and Poisson’s equation in spherical coordinates are encountered in a wide range mechanics). As might be expected the solutions of these equations have been well studied. The solutions to the angular part of the equation are the spherical harmonics . These are the appropriately normalized products of the associated Legendre polynomials in cos and the exp im Y m , 1 m 2 ℓ 1 4 m ! ℓ m ! P m cos  e im The Y m , satisfy L L Y l m ℓ ℓ 1 Y m and L z Y m mY m ;w h e r e L z z ̂ L The associate Legendre functions are as follows: P m x constant x 1 x 1 m /2 1 x 2 m F m , ℓ m 1, m 1; 1 x 2 F a , b , c ; z Γ 1 c Γ a Γ b n Γ a n Γ b n n ! Γ c n z n Γ a a 1 ! Some useful properties are: Y m , 1 m Y m , Y m ,2 1 Y m , The orthonormality condition: 0 2 0 Y m , Y m , sin d d ℓℓ mm Y m , Y m ℓℓ
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ℓ 0 m Y m ,
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This note was uploaded on 12/20/2010 for the course PHYS 402 taught by Professor J. during the Fall '09 term at Missouri S&T.

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Chapter5.p44.Greens.function - Boundary Value Problems in...

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