10. Compute the following residues. (d) Res (2n(1-2); 0), where n E N Hint: In (d), the result from Q21 of Problem Set

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10 Compute The Following Residues D Res 2n 1 2 0 Where N E N Hint In D The Result From Q21 Of Problem Set 1 (4.27 KiB) Viewed 19 times

Here is the result from Q21

10 Compute The Following Residues D Res 2n 1 2 0 Where N E N Hint In D The Result From Q21 Of Problem Set 2 (68.32 KiB) Viewed 19 times

10. Compute the following residues.
(d) Res (2n(1-2); 0), where n E N Hint: In (d), the result from Q21 of Problem Set 3 may be useful.
21. We prove by induction. For n = 1, since for every z € D(0; 1), the statement is true. 5+00 Lk=0 Now 1-z assume that the statement is true for n = m ≥ 1, i.e. 1 (1-z)m+1 -Ï (k + m) z* m for every z E D(0; 1). Then term-by-term differentiation gives +00 +00 m + 1 d 1 = Σ (k + m) kzk-² = [ (k + m + ¹1) (k + 1) z*, (1z)m+2 dz (1 − z)m+1 m k=0 for every z E D(0; 1), so +00 +00 k+1 (k+m+1)! k +1 1 (1-z)m+2 = [ (x + m + 1) k² + ¹1 ² ² = ) m m+1 m! (k+ 1)! m +1° k=0 k=0 +00 (k+ m + 1)! (m + 1)!k! ²-²₂k = Σ (k+h (k+m+1) m+1 k=0 k=0 for every z € D(0; 1), i.e. the statement is true for n = m +1 as well.