1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

56 CHAPTER 2 • COMPLEX FUNCTI ONS

y

• W=t+B
  u=x+a
  v=y+b

~-·· -·/B=~v+=i~t)

I I

I I

'----

F igure 2.5 The t ranslation w = T (z) = z + B = x +a+ i (y + b).

y v

p=r

ti>= 9+ a

u

Figure 2.6 The rotation w = R(z) = re•<s+").

If we let a be a fixed real number, then for z = r eie, the transformation

w = R (z) = ze^1 <> = re^1 1J ei°' = rei(IJ+a)

is a one-to-one mapping of the z plane onto thew plane and is called a rotation.

It can be visualized as a rigid rotation whereby the point z is rotated about

the origin through an angle a to its new position w = R(z). If we use polar

coordinates and designate w = pit/> in the w plane, then the inverse mapping is

This analysis shows that R is a one-to-one mapping of the z plane onto thew

plane. The effect of rotation is depicted in Figure 2.6.

• EXAMPLE 2.8 The ellipse centered at the origin with a horizontal major
  axis of four units and vertical minor axis of two units can be represented by the
  parametric equation

s (t) = 2 cost+ isin t = (2 cost, sin t), for 0 ~ t ~ 21T.