Page 408 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 16] FUNCTIONS OF A COMPLEX VARIABLE 399

Solved Problems

FUNCTIONS, LIMITS, CONTINUITY

16.1. Determine the locus represented by
(a) jz 2j¼ 3; ðbÞjz 2j¼jz þ 4j; ðcÞjz 3jþ jz þ 3j¼ 10.

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2
ðx 2Þ þ y ¼ 3or ðx 2Þ þ y ¼ 9, a circle with
(a) Method 1: jz 2j¼ jx þ iy 2j¼jx 2 þ iyj¼
center at ð2; 0Þ and radius 3.
Method 2: jz 2j is the distance between the complex numbers z ¼ x þ iy and 2 þ 0i.If thisdistance is
always 3, the locus is a circle of radius 3 with center at 2 þ 0i or ð2; 0Þ.
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2
(b) Method 1: jx þ iy 2j¼jx þ iy þ 4j or ðx 2Þ þ y ¼ ðx þ 4Þ þ y . Squaring, we find x ¼ 1, a
straight line.
Method 2: The locus is such that the distance from any point on it to ð2; 0Þ and ð 4; 0Þ are equal. Thus,
the locus is the perpendicular besector of the line joining ð2; 0Þ and ð 4; 0Þ,or x ¼ 1.
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2
2
2
(c) Method 2: The locus is given by ðx 3Þ þ y þ ðx þ 3Þ þ y ¼ 10 or ðx 3Þ þ y ¼ 10
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
2
ðx þ 3Þ þ y . Squaring and simplifying, 25 þ 3x ¼ 5 ðx þ 3Þ þ y . Squaring and simplifying
x 2 y 2
again yields þ ¼ 1, an ellipse with semi-major and semi-minor axes of lengths 5 and 4, respec-
25 16
tively.
Method 2: The locus is such that the sum of the distances from any point on it to ð3; 0Þ and ð 3; 0Þ is 10.
Thus the locus is an ellipse whose foci are at ð 3; 0Þ and ð3; 0Þ and whose major axis has length 10.
16.2. Determine the region in the z plane represented by each of the following.
(a) jzj < 1.
Interior of a circle of radius 1. See Fig. 16-3(a)below.
(b)1 < jz þ 2ij @ 2.
jz þ 2ij is the distance from z to 2i,sothat jz þ 2ij¼ 1isa circle of radius 1 with center at 2i,
i.e., ð0; 2Þ; and jz þ 2ij¼ 2isa circle of radius 2 with center at 2i. Then 1 < jz þ 2ij @ 2 represents
the region exterior to jz þ 2ij¼ 1 but interior to or on jz þ 2ij¼ 2. See Fig. 16-3(b)below.
(c) =3 @ arg z @ =2.
i
Note that arg z ¼ , where z ¼ e . The required region is the infinite region bounded by the lines
¼ =3 and ¼ =2, including these lines. See Fig. 16-3(c)below.

Fig. 16-3