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6. The addition theorem for spherical harmonics states that 4π 21+1 where P₁(cos y) = cos y = ', = sin 0 cos p + sin 0 sin oy + cos 02, and sin 0' cos d'a + sin e' sin o'y + cos 0'2. Σ Yim(0, φ)Yim(θ', φ), m=-1 = Rather than ask you to prove this, I just let you exercise the formula a little bit: (a) Show that cos y = sin sin ' cos(o- ') + cos 0 cos 0'. (b) What does the addition theorem mean when I = 0? This should be trivial. (c) What about when 1 = 1? Does this make sense?

6. The addition theorem for spherical harmonics states that 4π 21+1 where P₁(cos y) = cos y = ', = sin 0 cos p + sin 0 sin oy + cos 02, and sin 0' cos d'a + sin e' sin o'y + cos 0'2. Σ Yim(0, φ)Yim(θ', φ), m=-1 = Rather than ask you to prove this, I just let you exercise the formula a little bit: (a) Show that cos y = sin sin ' cos(o- ') + cos 0 cos 0'. (b) What does the addition theorem mean when I = 0? This should be trivial. (c) What about when 1 = 1? Does this make sense?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How to do question 6.
6. The addition theorem for spherical harmonics states that 4T Pi(cos y) = Yim (0, 0) Ym (8', o'), 21+1 where cos y = . ', m=-1 = sin cos p + sin 0 sin oy + cos 02, and = sin 0' cos d'a + sin 0' sin o'y + cos 0'2. Rather than ask you to prove this, I just let you exercise the formula a little bit: (a) Show that cos y sin sin ' cos(-o') + 'cos cos 0'. (b) What does the addition theorem mean when I = 0? This should be trivial. 1? Does this make sense? (c) What about when l = (d) What is Yim (0, 0)? For what m values is it non-zero? (e) What happens to the addition theorem when 0' = 0? S i 10 O X
Transcribed Image Text:6. The addition theorem for spherical harmonics states that 4T Pi(cos y) = Yim (0, 0) Ym (8', o'), 21+1 where cos y = . ', m=-1 = sin cos p + sin 0 sin oy + cos 02, and = sin 0' cos d'a + sin 0' sin o'y + cos 0'2. Rather than ask you to prove this, I just let you exercise the formula a little bit: (a) Show that cos y sin sin ' cos(-o') + 'cos cos 0'. (b) What does the addition theorem mean when I = 0? This should be trivial. 1? Does this make sense? (c) What about when l = (d) What is Yim (0, 0)? For what m values is it non-zero? (e) What happens to the addition theorem when 0' = 0? S i 10 O X
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