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Unformatted text preview: Problem Set 3 January 23, 2004 Due January 30, 2004 ACM 95b/100b 3pm in Firestone 303 E. Sterl Phinney (2 pts) Include grading section number 1. (4 5 points) The following (trivial once you get it) problem is designed to help those of you who had trouble with Problem Set 1s problem 7c. It will also help you find asymptotic expansions of Laplace integrals and their relatives. a) Evaluate Z a exp(- 1000 t ) dt (1) for a = 10- 4 , 10- 3 , 3 10- 3 , 9 10- 3 , 0.1, 10 and . Also give your answers in fixed decimal form to 7 digits [i.e. numbers like 0.1234567], and explain any trend you notice. b) Now consider the more general integral I ( a ) = Z a exp(- xt ) dt (2) for x 100. What range of values of a makes I ( a ) = I ( )(1 + ), with | | < . 01 (i.e. gives I ( ) to 1% accuracy)? Your answer may depend on x . c) Use reasoning motivated by part (b) to find the simplest function of x which approximates I ( x ) = Z e- xt (1 + t 2 ) dt (3) to 1% accuracy for x 100. Justify your error estimate. d) Use reasoning motivated by part (b) to find the simplest function of x which approximates I ( x ) = Z 1 / 5 e- xt p 2 t + t 3 / 4 + cos t dt (4) to 1% accuracy for x 100. Justify your error estimate. 2. (5 points) Prove that for an analytic function f ( t ) with Laplace transform F ( s ), lim s sF ( s ) = lim t 0+ f ( t ) . (5) 3. (6 4 points) Fun with Dirac a) Show that x d dx ( x ) =...
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