What does it mean that $$w=\frac{az+b}{cz+d}$$ can map a circle to a line, a line to a line, a circle to a circle or a line to a circle?
Thanks for your explanation. I noticed many problems regarding Conformal Mapping are asking to form a map in order to get a desired result, for instance, fix some points or map a area to another one. But I don't know how can I find the desired mapping. Is there a certain rules I should follow? The following is an example. Let U be the upper half plane from which the points of the