SOLUTIONS MANUAL COMPLEX ANALYSIS- A FIRST COURSE WITH APPLICATION ALL CHAPTERS INCLUDED 2023/2024

SOLUTIONS MANUAL COMPLEX ANALYSIS- A
FIRST COURSE WITH APPLICATION ALL
CHAPTERS INCLUDED 2023/2024

,SOLUTIONS MANUAL COMPLEX ANALYSIS- A FIRST
COURSE WITH APPLICATION ALL CHAPTERS INCLUDED
2023/2024




IV 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 X X X X X X X X X X X
2 X X X X X X X X
3 X X X X X X X
4 X X X X
5 X X X X
6 X X X X
7 X X X X X X X X X X X
8 X X X X X X X X




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,SOLUTIONS MANUAL COMPLEX ANALYSIS- A FIRST
COURSE WITH APPLICATION ALL CHAPTERS INCLUDED
2023/2024



I.1.1
Identify and sketch the set of points satisfying.
(a) |x — 1 — i| = 1 (f) 0 < Fm x < v
(b) 1 < |2x — 6| < 2 (g) —v < Re x < v
2 2
(c) |x — 1| + |x + 1| < 8 (h) |Re x| < |x|
(d) |x — 1| + |x + 1| ≤ 2 (i) Re (ix + 2) > 0
(e) |x — 1| < |x| (j) |x — i|2 + |x + i|2 < 2
Solution
Let x = x + iy, where x, y ∈ R.
(a) Circle, centre 1 + i, radius 1.

2 2 2
|x — 1 — i| = 1 e |(x — 1) + i (y — 1)| = 1 e (x — 1) + (y — 1) = 1

(b) Annulus with centre 3, inner radius 1/2, outer radius 1.


1 < |2x — 6| < 2 e 1 < 2 |x — 3| < 2 e 2 2 2 2
e 1/2 < |x — 3| < 1 e (1/2) < (x — 3) + y < 1
√
(c) Disk, centre 0, radius 3.

2
2
|x + iy — 1| + |x + iy + 1| < 8 e
2 2 2 2 2 2 √ 2
e (x — 1) + y + (x + 1) + y < 8 e x + y < 3

(d) Interval [—1, 1].

q q
|x — 1| + |x + 1| ≤ 2 e (x — 1) + 2
y2 ≤ 2 — (x + 1)2 + y2 e
2 2
q q
e (x — 1)2 + y2 ≤ 2 — (x + 1)22 + y2 e
2
q q
e (x + 1)2 + y2 ≤ x + 1 e (x + 1)2 + y2 ≤ (x + 1) e y = 0
Now, take y = 0 in the inequality, and compute the three intervals
2

, SOLUTIONS MANUAL COMPLEX ANALYSIS- A FIRST
COURSE WITH APPLICATION ALL CHAPTERS INCLUDED
2023/2024




x < —1, then |x — 1| + |x + 1| = — (x — 1) — (x + 1) = —2x ≥ 2,
—1 ≤ x ≤ 1 then |x — 1| + |x + 1| = — (x — 1) + (x + 1) = 2 ≤ 2
x > 1, then |x — 1| + |x + 1| = (x — 1) + (x + 1) = 2x ≥ 2.

(e) Half–plane x > 1/2.

2
2 2
|x — 1| < |x| e |x — 1| < |x| e |x + iy — 1| < |x + iy|2 e
2
e (x — 1) + y2 < x2 + y2 e x > 1/2

(f) Horizontal strip, 0 < y < v.
(g) Vertical strip, —v < x < v.
(h) s\R.
2 2 2 2 2
|Re x| < |x| e |Re (x + iy)| < |x + iy| e x < x + y e |y| > 0
(i) Half plane y < 2.

Re (ix + 2) > 0 e Re (i (x + iy) + 2) > 0 e —y + 2 > 0 e y < 2
(j) Empty set.


|x — i| 2 + |x + i|2 < 2 e |x + iy — i|2 + |x + iy + i|2 < 2 e
2 2
e x + (y — 1)2 + x2 + (y + 1)2 < 2 e x + y2 < 0




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