1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

58 CHAPTER 2 • COMPLEX FUNCT IONS

y

Ki

w=Kz

u=Kx

v

i .. v=Ky i

-+~~+-~~~-+--X U

K K

Figure 2.8 The magnificationw= S(z) = K z = Kx+i Ky.

Finally, if we let A = K ei" and B = a + ib , where K > 0 is a positive real

number, then the transformation

w = L (z) = A z + B

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

transformation. It can be considered as the composition of a rotation, a mag-

nification, and a translation. It has the effect of rotating the plane through an

angle given by a = Arg A, followed by a magnification by the factor K = IAI,

followed by a translation by the vector B = a+ ib. The inverse mapping is given

by z = L -^1 (w) = *w -~and shows that Lis a one-to-one mapping from the

z plane onto the w plane.

• EXAMPLE 2.9 Show that the linear transformation w = i z + i maps the
  right half-plane Re (z) ;::: 1 onto the upper half-plane Im (w);::: 2.

Solution (Method 1): Let A = {(x, y): x;::: 1}. To describe B =I (A), we

solve w = iz + i for z to get z = wii = -iw - 1 = 1-^1 (w). Using Equations

(2-5) and the method of Example 2.7 we have

u +iv= w = I (z) E B ¢===} 1-^1 (w) = z E A

¢===} -iw- 1 EA

¢===} v - 1 - iu E A

¢===} (v -1, -u) E A

¢=}v-1;:::1

¢===} v;::: 2.

Thus, B = {w = u +iv : v;::: 2 }, which is the same as saying Im (w);::: 2.