MATH
241S08

# Write the integrals to compute the coefficients in

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(Write the integrals to compute the coefficients in the final series expansions without attempting to evaluate them). 4. Draw pictures of the following subsets of the complex plane (give as much detail and be as accurate as possible): (a) { z : z = ¯ z } (b) { z : Im z > ( Re z ) 2 } 5. For each of the following sums say whether it is absolutely convergent or not, and give a justification for your answer: (a) n = 1 parenleftBig 2 2 + i parenrightBig n (b) n = 1 n i n + i 6. Give the radius of convergence of the following power series, justify your answer. (a) n = 1 n 4 z n (b) n = 1 z n ( n ! ) 2 7. Evaluate the following contour integral contintegraldisplay | z |= 4 dz z 2 4 contintegraldisplay | z + 2 |= 1 dz z 2 4 (2) 8. Evaluate the following contour integral contintegraldisplay | z |= 1 ( 2 z 2 + 3 ¯ z ) dz . (3)

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9. For the following function, locate and classify all the singularities as a pole of some finite order, essential singularity, or removeable singularity. Compute the residues at the poles of finite order: f ( z ) = e i z z + 1 z ( z 2 ) 2 . (4) 10. For the following functions give the radius of convergence of the Taylor series cen- tered at the indicated point (a) f ( z ) = e z 1 z centered as z 0 = 1 . (b) f ( z ) = ( z + 1 )( z 2 ) ( z 2 i ) 2 centered at z 0 = 2 . 11. For the function f ( z ) = 1 z ( z 4 ) , find the Laurent expansions valid in (a) 0 < | z | < 4 (b) 4 < | z | (c) 0 < | z 4 | < 4 (d) 4 < | z 4 | Extra Credit: Suppose that f ( z ) is an analytic function in the disk { z : | z 1 | < 2 } , and that f [ j ] ( 1 ) = 0 , for j = 0 , 1 , 2 , . . . What can we conclude about f ( z ) in this disk?
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