118Homework7

118Homework7 - e z about the point z = 1 6 Find the Laurent...

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Homework 7 Due Monday, Oct. 18 1. Prove that if z C and | z | < 1, then X n =0 z n = 1 1 - z . 2. Prove that if { z n } n =1 is a sequence of complex numbers and if one has n =1 z n = S , then it is the case that n =1 z n = S . 3. Find the Maclaurin series for f ( z ) = z z 4 + 9 and specify the largest circle centered at 0 for which it converges. (Hint: Use the result of Problem 1.) 4. Find the Taylor series expansion of cos z about the point z 0 = π/ 2. 5. Find the Taylor series expansion of
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Unformatted text preview: e z about the point z = 1. 6. Find the Laurent series for f ( z ) = z 2 sin ± 1 z 2 ² in the domain 0 < | z | < ∞ . 7. Consider the function f ( z ) = 1 z (1 + z 2 ) . (a) Find the Laurent series for f ( z ) in the region 0 < | z | < 1. (b) Find the Laurent series for f ( z ) in the region 1 < | z | < ∞ ....
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