Math 121A - Fall 2001 - Tokman - Final

Math 121A - Fall 2001 - Tokman - Final - 05/17/2002 FRI...

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Unformatted text preview: 05/17/2002 FRI 11:49 FAX 6434330 MOFFITT LIBRARY 001 Final Exam Math 121A (Section 2) — Fall 2001 M .Tol—cmctn Each problem counts 10 points Problem 4% 1. Solve the initial value problem using Laplace transform it” — 39’ + 2y = 1065‘, 11(0) : 1, y’(0) = 5- Problem # 2. The following periodic function f is defined over one period as f(ac):l~n:for0<rt<2. (a) Sketch several periods of flat) and expand it in an appropriate Fourier series. (1)) Write Parseval’s relation for the Fourier series of f Problem #73 3. Find out in which quadrants the roots of the following equation lie 23+z2+4z+9=0 Problem :fi’: 4. Show that if f (3;) is an odd function then its Fourier transform 9(a) is also an odd function. Problem # 5. Given yffl} u 2 f sin(y — Unit Find (%)y, (gfih and (3:14)” at 3: : 7r/2, y 2 7r. Problem # 6. Given f(:t) 2 er" + ln(2:r) (a) Find Maclaurin expansion of f (cc). (b) Find the interval of convergence of the Maclaurin series in (a) (including end points testsl). Justify your answer by mentioning the theorems / tests that you use to draw conclusions about convergence, and state explicitly if the convergence is abso- lute or conditional. 05/17/2002 FRI 11:49 FAX 6434330 MOFFITT LIBRARY 002 Problem # 7. Find the principal value of the integral 0° (reins; f _—dm. 0 95::2 w 7T2 Problem # 8. Sovc the following boundary value problem using Green's function y” ~l- 9y : sin 2:13. y(0) = 0, y(7r/2) z 0. Problem # 9. Given ~3/4 Hz) = ——#—w(z "1)2062 + 9). (a) Specify under what conditions and where on. a complex plane f(z) is analytic. b Identify all points where z is sin 'ular and specify the t pes of the sino'1_llal.‘ities. 8 Y o (c) Pick a branch of f(z) and compute residues of this branch at z 2 i, z : 3 and 2232'. Problem # 10. Let u(:r, t) satisfy the following equations 3n 821,5 “at Z is? i” u(:i:,0) = O (2) Mint) #9 O as I —+ ioo. (3) (a) Laplace transform the equation (1) and write the boundary conditions satisfied by the Laplace transform of u(:t, t). (b) Fourier transform the equation (1) and write the initial conditions satisfied by the Fourier transform of Mat, 1;). ...
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