s5_95-100a

# s5_95-100a - f ( z ) = 1 ( z-1)( z-2) (10 pts.) Obtain the...

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ACM95a/100a November 4, 2009 Problem Set V Please deposit your completed homework set to by the due-time in the Firestone 303 door-slot. Please write your Grading Section Number at the top of the ﬁrst page of your homework set. 1. (20 pts.) Let f ( z ) = k =1 k 3 ( z 3 ) k . Compute each of the following integrals giving justiﬁcation in each case (10 pts. each): ( i ) Z | z | =1 e iz f ( z ) dz ( ii ) Z | z | =1 f ( z ) z 4 dz 2. (15 pts.) Find the circle of convergence of the following series (5 pts. each). ( i ) X 1 n 2 n ( z - i ) n , ( ii ) X 0 n n z n , ( iii ) X 1 [3 + ( - 1) n ] n z n . 3. (15 pts.) Let f ( z ) = (1 + z ) α be the branch for which f (0) = 1. Find its Taylor series expansion about z = 0. What is the radius of convergence of the series ? ( α is an arbitrary complex number). 4. (25 pts.) Classify the singularities of
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Unformatted text preview: f ( z ) = 1 ( z-1)( z-2) (10 pts.) Obtain the Laurent expansion centered at z = 0 for each one of the following three regions: (a) | z | < 1, (5 pts.) (b) 1 < | z | < 2, (5 pts.) (c) | z | > 2 (5 pts.) [Hint: Re-express f using partial fractions.] 5. (25 pts.) Locate and classify all the singularities in the nite plane of f ( z ) = 1 z (exp ( z )-1) . (8 pts.) Calculate the residues at these singularities (8 pts.). Give the rst two terms of the Laurent expansion about z = 0 (8 pts.). Is the point at an isolated singularity? (1 pt.) Due at 5pm on Wed. October 28....
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## This note was uploaded on 01/14/2010 for the course ACM 1 taught by Professor Prof during the Fall '09 term at Caltech.

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