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Unformatted text preview: FINAL EXAM
Math 300, December 17, 2008 Last Name:
First Name: Student Number: Signature: The exam is worth a total of 100 points with duration 2.5 hours. No books, notes or calculators
are allowed. Justify all answers, show all work and explain your reasoning carefully. You will be graded on the clarity of your explanations as well as the correctness of your answers.
UBC Rules governing examinations: (1) Each candidate should be prepared to produce his/ her library/AMS card upon request. (2) N0 candidate shall be permitted to enter the examination room after the expiration of one half hour, or to
leave during the ﬁrst half hour of the examination. Candidates are not permitted to ask questions of the
invigilators, except in cases of supposed errors or ambiguities in the examination questions. (3) Candidates guilty of any of the following or similar practices shall be immediately dismissed from the
examination, and and shall be liable to disciplinary action: a) Making use of any books, papers or memoranda, other than those authorized by the examiners. b) Speaking or communicating with other candidates. 0) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness
will not be received. (4) Smoking is not permitted during examinations. Points
1 Problem 1 (10 points, 1 pt. each) For each of the following statements, circle T if the statement is
true, and F if the statement is false. T F If f (2) satisﬁes the Cauchy—Riemann equations at 20, then f (z) is differentiable at 20.
T F If f(z) has a pole at 20, then limz_,zO f(z) = 00. T F If f (z) is analytic in a domain D containing a simple closed contour F, then fF f (z) dz =
0. T F If the two power series 2,310 ak(z — z0)k and 22:0 bk(z — 2:0)" converge to the same
function in the disk {[2 —— 201 = 1}, then ak = bk for all k. T F There does not exist any function f (2) which is analytic at the point 0 and nonanalytic
everywhere else. T F The function L0g(z2) is analytic for all values of 2 except those on the negative real axis.
T F Any entire function is the complex derivative of another entire function. T F If f(z) has an essential singularity at 20, then Res((z — zo)f(z); 2:0) : 0. T F If f(z) and 9(2) have simple poles at 0, then (fg)(z) has a simple pole at 0. T F If the disk of convergence of the Taylor series of a function f (z) is {Izl = 2}, then the
disk of convergence for the Taylor series of f (22) is {lzl = 4}. Problem 2 (10 points) Use the LIMIT DEFINITION of differentiability to show that for any 30 75 07 f (z) z 22 is not differentiable at 20. Problem 3 (10 points) Suppose that the functions f (z) and 9(2) are entire,  f (2)1 < g(z)] for all z,
and f(0) 74 0. Show that f(z) has no zeros. (In other words, ShOW that f(z) 7Q 0 for all z.) Problem 4 (12 points) Find
1 —d2,
r2: where F is the portion of the circle {z = ﬂ}, traversed counterclockwise, beginning at —1 + z',
and ending at l + 2'. 4+1: [+42 Problem 5 (8 points) Let f (z) be a function analytic on the disk {z < 1}. Show that it cannot be
true that If (k)(0)l Z kl2k for all nonnegative integers k. Problem 6 (12 points) Find the Laurent series centered at 20 = 0 of in the following domains. (3) D = {IZI < 3}
(b) D = {Z > 3} Problem 7 (12 points) Classify all isolated singularities of M = —Z (Z ‘ 1) sin2(7rz) I Problem 8 (13 points) Find eTfZ where C is the negatively oriented circle {lzl = 2}. dz, Problem 9 (13 points) Using RESIDUE THEORY, evaluate v /°° $2+x dx
p. —oo(.’L'2+1)2 . (In other words, ﬁnd limp.”O ffp (76$ dx.) 1n ...
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 Winter '10
 Dr.AlejandroCortas
 Math

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