1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.2 • THE MAPPINGS w =Zn AND w = zfi 63

(b) Show, a.dditionally, that if g is on~to-one from B onto S, then h ( z) is

one-to-one from A onto S, where h (z) = f (g (z)).

12. Letw = f(z) = (3+4i}z-2+i.

(a) Find the image of the disk lz - l l < 1.

(b) Find the image of the line x = t, y = 1 -2t for -oo < t < oo.

(c) Find the image of the half-plane Im ( z) > 1.

( d) For parts a and b, and c, sketch the mapping, identify the points z 1 = 0,

z2 = 1 -i, and z3 = 2, and indicate their images.

13. Let w = (2 + i} z - 2i. Find the triangle onto which the triangle with vertices
    z 1 = -2 + i, z2 = -2 + 2i, and zs = 2 + i is mapped.
    1 4. Let S (z) = K z, where K > 0 is a positive real constant. Show that the equation
    IS (zi) - S (z2)1 = K lz1 - z2I holds and interpret this result geometrically.


14. Find the linear transfor mations w = f (z) t hat satisfy the following condit ions.

(a) The points z 1 = 2 and z 2 = - 3i map onto w 1 = 1 +i and Wz = 1.

(b) T he circle lzl = 1 maps onto the circle lw - 3 + 2il = 5, and f (-i) =

3 +3i.

( c) The triangle with vertices - 4 + 2i, - 4 + 7i, and 1 + 2i maps onto the

triangle with vertices 1, 0, and 1 + i, respectively.

16. Give a proof that the image of a circle under a linear transformation is a circle.
    Hint: Let the circle have the parametrization z = zo + Re", 0 :s; t :s; 27r.


17. Prove that the composition of two linear transformations is a linear transformation.


18. Show that a linear t ransformation that maps the circle lz - zol = R 1 onto the

circle lw -wol = Rz can be expressed in the form

A (w -wo) R1 = (z - zo) R2, where IAI = 1.

2. 2

1

THE MAPPINGS w = zn and w = zn

In this section we turn our attention to power functions.

For z = re^18 'I 0, we can express the function w = f (z) = z^2 in polar

coordinates as

If we also use polar coordinates for w = pe•<P in the w plane, we can express this
mapping by the system of equations

p = r^2 and </> = 29.

Because an argument of the product (z) (z} is twice an argument of z, we
say that f doubles angles at the origin. Points that lie on the ray r > 0, 9 = a