homework6 - z for the function f ( z ) = 1 z 2 (1-z ) and...

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Due 02-27-09 Physics 132 Winter 2009 P. Kraus HOMEWORK #6 Material covered: sections 55-57, 59, 60, 62, 68-70. Hand into box marked “132” outside 1-707D PAB by 3pm Fri. —————————————————————————————————————————— 1) Check the convergence of the Taylor series 1 1 - z = n =0 z n . Define the partial sum S N = N n =0 z n . For z = (1+ i ) / 2, check that | S N | approaches | 1 1 - z | as N increases. Do this by finding how big N needs to be in order to get 5% accuracy? Use this value of N but now for z = (1+ i ), and compute | S N | and | 1 1 - z | . Why is the accuracy so bad in this case? Note: You are encouraged to use a computer/calculator for this problem! 2) BC p. 196 problem 7: Derive the Taylor series representation 1 1 - z = X n =0 ( z - i ) n (1 - i ) n +1 , ( | z - i | < 2) 3) BC p. 197 problem 10: What is the largest circle within which the Maclaurin series for the function tanh z converges to tanh z ? Write the first two nonzero terms of that series. 4) BC p. 206 problem 4: Give two Laurent series expansions in powers of
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Unformatted text preview: z for the function f ( z ) = 1 z 2 (1-z ) and specify the regions in which these expansions are valid Ans : X n =0 z n + 1 z + 1 z 2 , (0 < | z | < 1);- X n =3 1 z n , (1 < | z | < ) 5) BC p. 206 problem 6: Show that when 0 < | z-1 | < 2, z ( z-1)( z-3) =-3 X n =0 ( z-1) n 2 n +2-1 2( z-1) 6) BC p. 239 problem 1: Find the residues at z = 0 of the functions ( a ) 1 z + z 2 ; ( b ) z cos 1 z ; ( c ) z-sin z z ; ( d ) cot z z 4 ; ( e ) sinh z z 4 (1-z 2 ) Ans ( a ) 1; ( b )-1 / 2; ( c ) 0; ( d )-1 / 45; ( e ) 7 / 6 7) BC p. 239 problem 2: Use Cauchys residue theorem to evaluate the integral of each of these functions around the circle | z | = 3 in the positive sense (counterclockwise contour) ( a ) e-z z 2 ; ( b ) e-z ( z-1) 2 ; ( c ) z 2 e 1 /z ; ( d ) z + 1 z 2-2 z Ans ( a )-2 i ; ( b )-2 i/e ; ( c ) i/ 3; ( d ) 2 i 1...
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This note was uploaded on 03/30/2010 for the course PHYSICS 132 taught by Professor Staff during the Winter '06 term at UCLA.

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