The most general type of transformation between one Argand diagram, in the
z-plane, and another, in the
Z-plane, that gives one and only one value of
Z for each value of z (and conversely) is known as the general bilinear transformation and takes the form:
$$z=\frac{aZ+b}{cZ+d}$$
a)Confirm that the transformation from the
Z-plane to the z-plane is also a general bilinear transformation.

b)Recalling that the equation of a circle can be written in the form
$$λ=\left| \frac{z-z1}{z-z2} \right|$$,$$ λ\neq1$$
show that the general bilinear transformation transforms circles into circles
(or straight lines). What is the condition that z1,z2 and λ
must satisfy if the transformed circle is to be a straight line?

Question

The most general type of transformation between one Argand diagram, in the
z-plane, and another, in the
Z-plane, that gives one and only one value of
Z for each value of z (and conversely) is known as the general bilinear transformation and takes the form:
$$z=\frac{aZ+b}{cZ+d}$$
a)Confirm that the transformation from the
Z-plane to the z-plane is also a general bilinear transformation.

b)Recalling that the equation of a circle can be written in the form
$$λ=\left| \frac{z-z1}{z-z2} \right|$$,$$ λ\neq1$$
show that the general bilinear transformation transforms circles into circles
(or straight lines). What is the condition that z1,z2 and λ
must satisfy if the transformed circle is to be a straight line?