Math 121A - Fall 2001 - Tokman - Final

Math 121A - Fall 2001 - Tokman - Final - FRI 11:49 FAX...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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;). ...
View Full Document

This note was uploaded on 11/21/2009 for the course MATH 121a taught by Professor Staff during the Fall '08 term at Berkeley.

Ask a homework question - tutors are online