1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.1 • FUNCTIONS AND LINEAR MAPPINGS 57

• 1
  (a)

2

1.5

I

0.5

0 ~::::'.::~.......+--

0.5 I 1.5 2 2.5 3 3.5

(b)

Figure 2.7 (a) Plot of the original ellipse; (b} plot of the rotated ellipse.

Suppose that we wanted to rotate the ellipse by an angle of i radians and shift

the center of the ellipse 2 units to the right and 1 unit up. Using complex
arithmetic, we can easily generate a parametric equation r ( t) that does so:

r(t) = s(t)e'~ + (2+i)

= (2cost + isint) (cos~+ isin ~) + (2 + i)

= (2costcos ~ - sintsin i) + i (2costsin i + sintcos i) + (2 + i)

= ( v'3 cost - ~sin t + 2) + i (cost + v; sin t + 1)

= ( v'3 cost - ~sin t + 2, cost + v; sin t + 1) , for 0 ~ t ~ 271".

Figure 2.7 shows parametric plots of these ellipses, using the software program

Maple.

If we let K > 0 be a fixed positive real number, then the transformation

w = S(z) = Kz = Kx+iKy

is a one-to-one mapping of the z plane onto the w plane and is called a magni-

fication. If K > 1, it has the effect of stretching the distance between points

by the factor K. If K < 1, then it reduces the distance between points by the

factor K. The inverse transformation is given by

z= S- 1( w ) = -w=^1 -u+i-v^1. 1

K K K

and shows that S is one-to-one mapping from the z plane onto the w plane. The

effect of magnification is shown in Figure 2.8.