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Chem Differential Eq HW Solutions Fall 2011 83

# Chem Differential Eq HW Solutions Fall 2011 83 - Section...

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Section 5.3 Spherical Harmonics and the General Dirichlet Problem 83 Solutions to Exercises 5.3 1. (c) Starting with (4) with n = 2, we have Y 2 ,m ( θ,φ ) = 5 4 π (2 - m )! (2 + m )! P m 2 (cos θ ) e imφ , where m = - 2 , - 1 , 0 , 1 , 2. To compute the spherical harmonics explicitely, we will need the explicit formula for the associated Legendre functions from Example 1, Section 5.7. We have P - 2 2 ( x ) = 1 8 (1 - x 2 ); P - 1 2 ( x ) = 1 2 x 1 - x 2 ; P 0 2 ( x ) = P 2 ( x ) = 3 x 2 - 1 2 ; P 1 2 ( x ) = - 3 x 1 - x 2 ; P 2 2 ( x ) = 3(1 - x 2 ) . So Y 2 , - 2 ( θ,φ ) = 5 4 π (2 + 2)! (2 - 2)! P - 2 2 (cos θ ) e - 2 = 5 4 π 4! 1 1 8 =sin 2 θ (1 - cos 2 θ ) e - 2 = 30 π 1 8 sin 2 θe - 2 = 3 4 5 6 π sin 2 θe - 2 ; Y 2 , - 1 ( θ,φ ) = 5 4 π (2 + 1)! (2 - 1)! P - 1 2 (cos θ ) e - = 5 4 π 3! 1! 1 2 cos θ =sin θ 1 - cos 2 θe - = 15 2 π 1 2 cos θ sin θe - = 3 2 5 6 π cos θ sin θe - . Note that since 0 θ π , we have sin θ 0, and so the equality 1 - cos 2 θ = sin θ that we used above does hold. Continuing the list of spherical harmonics, we have
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