Page 881 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 861

Section 19.4 The Complex Logarithm
1. ln(4) + i(3π/2 + 2nπ), with n any integer.
√
3. ln(5) + (2n + 1)πi 5. 1 ln 85 + (arctan(−2/9) + (2n + 1)π)i
2
Section 19.5 Powers
1. ie −(π/2+2nπ) 3. e −(π/2+2nπ)
√ √
5. e 9π/4+6nπ cos 3ln 2 2 − i sin 3ln 2 2
7. e i(π+4nπ)/8 , n = 0,1,2,3 9. 16e (2n+1)π [cos(ln(4)) − i sin(ln(4))]
11. 2e (2n+1)πi/4 , n = 0,1,2,3
13. e nπi/3 , n = 0,1,2,3,4,5

CHAPTER TWENTY COMPLEX INTEGRATION
Section 20.1 The Integral of a Complex Function
1. 8 − 2i 3. 3 (1 + i) 5. − 13 + 2i
2 2
1
7. − [cosh(8) − cosh(2)] 9. −cos(2)sinh(1) − i sin(2)cosh(1)
2
11. 10 + 210i 13. 25i/2 15. 2 3 (1 + i)
√
17. One bound is 1/ 2. Any larger number is also a bound.
Section 20.2 Cauchy’s Theorem
1. 0 by Cauchy’s theorem 3. 0 5. 2πi
7. 0 9. 0 11. 4πi

Section 20.3 Consequences of Cauchy’s Theorem
2
1. 32πi 3. 2πi(−8 + 7i) 5. −2πe [cos(1) − sin(1)i]
7. πi[6cos(12) − 36sin(12)] 9. −512π(1 − 2i)cos(256)
13
11. − − 39i 13. 2π
2
CHAPTER TWENTY ONE SERIES REPRESENTATIONS OF FUNCTIONS
Section 21.1 Power Series
1. Radius 2, open disk |z + 3i| < 2 3. 1/e, |z − 1 + 3i| < 1/e
5. 2,|z + 8i| < 2 7. No (zero is further from 2i than i is)
∞ (−1) n
2n 2n

9. 2 z for |z| < ∞
n=0
(2n)!
11. −3 + (1 − 2i)(z − 2 + i) + (z − 2 + i) 2
2
13. z − 9) = 63 − 16i + (−16 + 2i)(z − 1 − i) + (z − 1 − i) 2
3 2i 6 4i
3
4
2
15. 1 + iz + z + z + z + z 5
2 3! 4! 5!
17. First expand e zw in a Maclaurin series to write
∞
1 z n 1 z n+k w k−n−1
zw
e dw = dw.
2πi γ n!w n+1 2πi n!k!
γ
k=0
θ
Parametrize w = e for 0 ≤ θ ≤ 2π in this integral to obtain
1 z n (z )
n 2
zw
e dw = .
2πi γ n!w n+1 (n!) 2
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October 14, 2010 17:50 THM/NEIL Page-861 27410_25_Ans_p801-866