MA530_JAN94 - 3. Suppose there is R > 1 so that...

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QUALIFYING EXAMINATION JANUARY 1994 MATH 530 Please answer each question on a separate sheet of paper! 1. (a) The function f ( z )= 4 (1 + z )(3 - z ) has Laurent series (I) k =0 ± 1+ ( - 1) k 3 k +1 ² ( z - 2) k (II) - 1 k = -∞ ( - 1+( - 3) - ( k +1) ) ( z - 2) k (III) - 1 k = -∞ - ( z - 2) k + k =0 ( - 1) k 3 k +1 ( z - 2) k Find the sets of absolute convergence for each of these series. (b) Suppose the function f is analytic in the plane except for simple poles at z = - 1and z =3 and has Laurent series (I) X k =0 a k ( z - 2) k (II) - 1 X k = -∞ b k ( z - 2) k (III) X k = -∞ c k ( z - 2) k Letting Γ = { z : | z - 3 | =1 } oriented counterclockwise, express the integral Z Γ f ( z ) dz in terms of the coefficients of the Laurent series above, and justify your answer. 2. The function g is analytic in the plane except for four poles, including poles at - 1, 2, and 3 + 4 i . Moreover, g is real–valued on the interval { z :Im( z )=0 , - 1 <z< 2 } of the real axis. (a) Prove that g is real–valued on the whole real axis except for its poles. (b) Find the location of the fourth pole and justify your answer.
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Unformatted text preview: 3. Suppose there is R &gt; 1 so that h ( z ) is analytic in the disk { z : | z | &lt; R } . Prove that if | h ( z ) | 1 for | z | 1 and h (0) = 0 and h (1) = 1, then | h (1) | 1. (Hint: you may wish to consider lim r 1-( h (1)-h ( r )) / (1-r ).) 4. (a) Express the arctangent function in terms of the logarithm. (b) Let A ( z ) be the branch of the arctangent function that is analytic except for { z : Re( z ) = 0 , | Im( z ) | 1 } and that has A (0) = . Find, justifying your work, lim t + Re( A ( t + i )) (where, as usual, t + means t is positive and real as it approaches 0). 5. Use the residue theorem to evaluate Z t 4 + t 4 dt Justify your answer by careful statements of your contours and estimates....
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