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Mathematic Methods HW Solutions 66

Mathematic Methods HW Solutions 66 - Chapter 14 3.1 3.5 3.9...

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Chapter 14 66 3.1 1 2 + i 3.2 - (2 + i ) / 3 3.3 0 3.4 iπ/ 2 3.5 - 1 3.6 - 1, - 1 3.7 π (1 - i ) / 8 3.8 i/ 2 3.9 1 3.10 (2 i - 1) e 2 i 3.11 2 πi 3.12 (a) 5 3 (1 + 2 i ) (b) 1 3 (8 i + 13) 3.16 0 3.17 (a) 0 (b) 3.18 3 / 6 3.19 16 3.20 (a) 0 (b) - 17 πi/ 4 3.22 - 3 / 108 3.23 72 3.24 - 17 iπ/ 96 4.3 For 0 < | z | < 1: 1 2 z - 1 + 3 4 + 7 8 z + 15 16 z 2 · · · ; R (0) = 1 2 For 1 < | z | < 2: - ( · · · z - 4 + z - 3 + z - 2 + 1 2 z - 1 + 1 4 + 1 8 z + 1 16 z 2 + 1 32 z 3 · · · ) For | z | > 2: z - 3 + 3 z - 4 + 7 z - 5 + 15 z - 6 · · · 4.4 For 0 < | z | < 1: - 1 4 z - 1 - 1 2 - 11 16 z - 13 16 z 2 · · · ; R (0) = - 1 4 For 1 < | z | < 2: · · · + z - 3 + z - 2 + 3 4 z - 1 + 1 2 + 5 16 z + 3 16 z 2 · · · For | z | > 2: z - 4 + 5 z - 5 + 17 z - 6 + 49 z - 7 · · · 4.5 For 0 < | z | < 2: 1 2 z - 3 - 1 4 z - 2 - 1 8 z - 1 - 1 16 - 1 32 z - 1 64 z 2 ; R (0) = - 1 8 For | z | > 2: z - 3 + z - 4 + 2 z - 5 + 4 z - 6 + 8 z - 7 · · · 4.6 For 0 < | z | < 1: z - 2 - 2 z - 1 + 3 - 4 z + 5 z 2 · · · ; R (0) = - 2 For | z | > 1: z - 4 - 2 z - 5 + 3 z - 6 · · · 4.7 For | z | < 1: 2 - z + 2 z 2 - z 3 + 2 z 4 - z 5 · · · ; R (0) = 0 For | z | > 1: z - 1 - 2 z - 2 + z - 3 - 2 z - 4 · · · 4.8 For | z | < 1: - 5 + 25 6 z - 175 36 z 2 · · · ; R (0) = 0 For 1 < | z | < 2: - 5( · · · + z - 3 - z - 2 + z - 1 + 1 6 z + 1 36 z 2 + 7 216 z 3 · · · ) For 2 < | z | < 3: · · · + 3 z - 3 + 9 z - 2 - 3 z - 1 + 1 - 1 3 z + 1 9 z 2 - 1 27 z 3 · · · For | z | > 3: 30( z - 3 - 2 z - 4 + 9 z - 5 · · · ) 4.9 (a) regular (b) pole of order 3 (c) pole of order 2 (d) pole of order 1 4.10 (a) simple pole (b) pole of order 2 (c) pole of order 2 (d) essential singularity 4.11 (a) regular (b) pole of order 2 (c) simple pole (d) pole of order 3 4.12 (a) pole of order 3 (b) pole of order 2 (c) essential singularity (d) pole of order 1 6.1 z - 1 - 1 + z - z 2 · · · ; R = 1 6.2 ( z - 1) - 1 - 1 + (
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