In many applications that involve boundary-value problems associated with Laplace’s equation, it is necessary to find a conformal mapping that maps a disk onto the half-plane v ≥ 0. Such a mapping would have to map the circular boundary of the disk to the boundary line of the half-plane. An important class of elementary conformal mappings that map circles to lines (and vice versa) are the fractional transformations. In this section we will define and study this special class of mappings.
Linear Fractional Transformation
If a, b, c, and d are complex constants with ad − bc ≠ 0, then the complex ...