math121b:midterm2
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math121b:midterm2 [2020/04/10 19:23] pzhou created |
math121b:midterm2 [2020/04/19 23:06] (current) pzhou [3. Bessel Function (30 points)] |
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| 1. Compute $P_3(x)$ using Rodrigue formula. (5 points) | 1. Compute $P_3(x)$ using Rodrigue formula. (5 points) | ||
| - | 2. Prove the recursion relation 5.8(c) (10 points) | + | 2. Prove the recursion relation 5.8( c) (10 points) |
| $$ P_l' | $$ P_l' | ||
| using the generating function | using the generating function | ||
| $$\Phi(x,h) = \sum_{n=0}^\infty h^n P_n(x) = \frac{1}{\sqrt{1 - 2 x h + h^2} } $$ | $$\Phi(x,h) = \sum_{n=0}^\infty h^n P_n(x) = \frac{1}{\sqrt{1 - 2 x h + h^2} } $$ | ||
| - | 3. Compute $\int_{-1}^1 x^n P_n(x) dx$ in the following steps: | + | 3. Compute $\int_{-1}^1 x^n P_n(x) dx$. ( 10 points) |
| + | |||
| + | Hint:( you don't have to use these hints) | ||
| * Find the constant $c$, such that $P_n(x) = c x^n + \z{ lower order terms}$. (try Rodrigue formula) | * Find the constant $c$, such that $P_n(x) = c x^n + \z{ lower order terms}$. (try Rodrigue formula) | ||
| * Show that $\int_{-1}^1 P_n(x)^2 | * Show that $\int_{-1}^1 P_n(x)^2 | ||
| + | * Look up $\int_{-1}^1 P_n(x)^2 | ||
| Line 36: | Line 39: | ||
| 1. Problem 12.1. (7 points) | 1. Problem 12.1. (7 points) | ||
| - | 2. Problem 15.6 (7 points) | + | 2. Problem 15.6 (7 points) |
| 3. Problem 19.1 (10 points) | 3. Problem 19.1 (10 points) | ||
| 4. Problem 20.3, 20.6, 20.7 (6 points) | 4. Problem 20.3, 20.6, 20.7 (6 points) | ||
| + | * 20.3: $ 4/\pi$ | ||
| + | * 20.6: $-1/ | ||
| + | * 20.7: $(1/x) e^{i (x - (n+1) \pi /2)} $ | ||
| ===== 4. Solving PDE with separation of variables (30 points) ===== | ===== 4. Solving PDE with separation of variables (30 points) ===== | ||
| Line 60: | Line 66: | ||
| $$ u(0, \theta) = \cos^2(\theta). $$ | $$ u(0, \theta) = \cos^2(\theta). $$ | ||
| + | Hint: you may find the following formula useful | ||
| + | $$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2 \cos^2 \theta - 1. $$ | ||
math121b/midterm2.1586571827.txt.gz · Last modified: 2020/04/10 19:23 by pzhou
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