FUNCTIONS OF A COMPLEX VARIABLE in VS .NET

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FUNCTIONS OF A COMPLEX VARIABLE
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FUNCTIONS, LIMITS, CONTINUITY 16.1. Determine the locus represented by (a) jz 2j 3; b jz 2j jz 4j; c jz 3j jz 3j 10.
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q x 2 2 y2 3 or x 2 2 y2 9, a circle with
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(a) Method 1: jz 2j jx iy 2j jx 2 iyj center at 2; 0 and radius 3.
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Method 2: jz 2j is the distance between the complex numbers z x iy and 2 0i. If this distance is always 3, the locus is a circle of radius 3 with center at 2 0i or 2; 0 . q q 2 (b) Method 1: jx iy 2j jx iy 4j or x 2 2 y2 x 4 2 y . Squaring, we nd x 1, a straight line. Method 2: The locus is such that the distance from any point on it to 2; 0 and 4; 0 are equal. Thus, the locus is the perpendicular besector of the line joining 2; 0 and 4; 0 , or x 1. q q q Method 2: The locus is given by x 3 2 y2 x 3 2 y2 10 or x 3 2 y2 10 q q x 3 2 y2 . Squaring and simplifying, 25 3x 5 x 3 2 y2 . Squaring and simplifying x2 y2 again yields 1, an ellipse with semi-major and semi-minor axes of lengths 5 and 4, respec25 16 tively. Method 2: The locus is such that the sum of the distances from any point on it to 3; 0 and 3; 0 is 10. Thus the locus is an ellipse whose foci are at 3; 0 and 3; 0 and whose major axis has length 10.
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16.2. Determine the region in the z plane represented by each of the following. (a) jzj < 1.
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Interior of a circle of radius 1. See Fig. 16-3(a) below.
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(b) 1 < jz 2ij @ 2.
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jz 2ij is the distance from z to 2i, so that jz 2ij 1 is a circle of radius 1 with center at 2i, i.e., 0; 2 ; and jz 2ij 2 is a circle of radius 2 with center at 2i. Then 1 < jz 2ij @ 2 represents the region exterior to jz 2ij 1 but interior to or on jz 2ij 2. See Fig. 16-3(b) below.
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(c) =3 @ arg z @ =2.
Note that arg z , where z ei . The required region is the in nite region bounded by the lines  =3 and  =2, including these lines. See Fig. 16-3(c) below.
Fig. 16-3
FUNCTIONS OF A COMPLEX VARIABLE
[CHAP. 16
16.3. Express each function in the form u x; y iv x; y , where u and v are real: (a) z3 ; b 1= 1 z ; c e3z ; d ln z.
a w z3 x iy 3 x3 3x2 iy 3x iy 2 iy 3 x3 3ix2 y 3xy2 iy2 x3 3xy2 i 3x2 y y3 Then u x; y x3 3xy2 ; v x; y 3x2 y y3 . b w 1 1 1 1 x iy 1 x iy 1 z 1 x iy 1 x iy 1 x iy 1 x 2 y2 Then u x; y c d 1 x y ; v x; y : 1 x 2 y2 1 x 2 y2 and u e3x cos 3y; v e3x sin 3y
e3z e3 x iy e3x e3iy e3x cos 3y i sin 3y
q ln z ln ei ln  i ln x2 y2 i tan 1 y=x and u 1 ln x2 y2 ; 2 v tan 1 y=x
Note that ln z is a multiple-valued function (in this case it is in nitely many-valued), since  can be increased by any multiple of 2. The principal value of the logarithm is de ned as that value for which 0 @  < 2 and is called the principal branch of ln z.
16.4. Prove
(a) sin x iy sin x cosh y i cos x sinh y (b) cos x iy cos x cosh y i sin x sinh y.
eiz e iz ; 2i eiz e iz 2
We use the relations eix cos z i sin z; e ix cos z i sin z, from which sin z cos z
Then sin z sin x iy
 y   y  1 y e e y e e y fe cos x i sin x e y cos x i sin x g sin x i cos x 2i 2 2
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