week3-hw - R C fdz when 1 f z = x 2 iy 2 C = z t = t 2 it |...

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Find the region of convergence of the following series: 1. (for grade) X n =1 ( n + ( - 1) n 2 n ) z n 2. X n =1 ( - 1) n n z n 3. X n =1 1 n 3 / 2 z n The geometric series: n =1 z n = 1 1 - z converges absolutely for | z | < 1. 1. Find the region of convergence of X n =1 n 2 z n , 2. (for grade) Find explicitly the function to which this power series is con- vergent in the region of convergence. 3. Describe the region where this function is defined. Evaluate the integral
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Unformatted text preview: R C fdz , when : 1. f ( z ) = x 2 + iy 2 , C = { z ( t ) = t 2 + it | ≤ t ≤ 1 } , 2. (for grade) f ( z ) = 1 z , C = { z ( t ) = sin t + i cos t | ≤ t ≤ 2 π } , 3. f ( z ) = 1 z +2 , C = { z ( t ) = cos t + i sin t | ≤ t 2 π } , 4. (for grade) f ( z ) = 1 z ( z +1)( z +2) , C = { z ( t ) = t + 1 | ≤ t < ∞} (improper integral). 1...
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