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math121b:midterm2 [2020/04/11 18:56]
pzhou [2. Legendre Function (25 points)] 		math121b:midterm2 [2020/04/19 23:06] (current)
pzhou [3. Bessel Function (30 points)]
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 1. Compute P3(x)P_3(x) using Rodrigue formula. (5 points) 1. Compute P3(x)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'(x) - xP_{l-1}'(x) = l P_{l-1}(x)   P_l'(x) - xP_{l-1}'(x) = l P_{l-1}(x)
 using the generating function  using the generating function 
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   * Find the constant cc, such that Pn(x)=cxn+ lower order termsP_n(x) = c x^n + \z{ lower order terms}. (try Rodrigue formula)   * Find the constant cc, such that Pn(x)=cxn+ lower order termsP_n(x) = c x^n + \z{ lower order terms}. (try Rodrigue formula)
   * Show that $\int_{-1}^1 P_n(x)^2  dx = \int_{-1}^1 c x^n P_n(x) dx$   * Show that $\int_{-1}^1 P_n(x)^2  dx = \int_{-1}^1 c x^n P_n(x) dx$
 +  * Look up $\int_{-1}^1 P_n(x)^2  dx$ in Boas. 
  
  
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 1. Problem 12.1. (7 points) 1. Problem 12.1. (7 points)
  
-2. Problem 15.6 (7 points)+2. Problem 15.6 (7 points) Making the computer plot is optional. 
  
 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/π 4/\pi
 +  * 20.6: $-1/(2n+1)$. 
 +  * 20.7: (1/x)ei(x−(n+1)π/2)(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) =====
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 u(0,θ)=cos⁡2(θ). u(0, \theta) = \cos^2(\theta).  u(0,θ)=cos⁡2(θ). u(0, \theta) = \cos^2(\theta).
  
 +Hint: you may find the following formula useful 
 +cos⁡(2θ)=cos⁡2(θ)−sin⁡2(θ)=2cos⁡2θ−1.\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2 \cos^2 \theta - 1.
  
  
math121b/midterm2.1586656606.txt.gz · Last modified: 2020/04/11 18:56 by pzhou

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