Chem Differential Eq HW Solutions Fall 2011 85

Chem Differential Eq HW Solutions Fall 2011 85 - Section...

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Section 5.3 Spherical Harmonics and the General Dirichlet Problem 85 where we used ± 1 - 1 P 1 ( x ) dx = 0, because P 1 ( x )= x is odd. Using n = 1 and m = - 1 in (9), and appealing to the formulas for the associated Legendre functions from Section 5.7, we get A 1 , - 1 = 1 2 π ² 3 4 π 2! 0! = 2 π ( - 1) i ³ ´µ · 2 π 0 φe · π 0 P - 1 1 (cos θ ) sin θdθ = - i ² 3 2 π 1 2 = π 2 ³ ´µ · π 0 sin 2 θdθ ( P - 1 1 (cos θ )= 1 2 sin θ ) = - i 4 ² 3 π 2 . Using n = 1 and m = 1 in (9), and appealing to the formulas for the associated Legendre functions from Section 5.7, we get A 1 , 1 = 1 2 π ² 3 4 π 0! 2! = 2 π 1 i ³ ´µ · 2 π 0 φe - · π 0 P 1 1 (cos θ ) sin θdθ = i ² 3 8 π = - π 2 ³ ´µ · π 0 - sin 2 θdθ ( P 1 1 (cos θ )= - sin θ ) = - i 4 ² 3 π 2 . (c) The formula for A n, 0 contains the integral ± π 0 P 0 n (cos θ ) sin θdθ . But
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