Question

what will be the image of the half plane x>c where c>0 under the transformation w=1/z

Answer 1

Answer:

The Transformation w=1/z

Consider the equation

w=1z

which establishes a one to one correspondence between the nonzero points of the z and w planes. Since zz¯¯¯=|z|2, the mapping can be described by means of the successive transformations

g(z)=z|z|2,f(z)=g(z)¯¯¯¯¯¯¯¯¯.

The first transformation g(z) is an inversion with respect to the unit circle |z|=1. That is, the image of a nonzero point z is the point g(z) with the properties

|g(z)|=1|z|andarg g(z)=arg z.

Thus the points exterior to the circle |z|=1 are mapped onto the nonzero points interior to it, and conversely. Any point on the circle is mapped onto itself. The second transformation f(z)=g(z)¯¯¯¯¯¯¯¯¯ is simply a reflection in the real axis.