Introduction

Mobius Transformations

Another important class of elementary mappings was studied by Augustus Ferdinand Mobius (1790-1868). These mappings are conveniently expressed as the quotient of two linear expressions and are commonly known as linear fractional or bilinear transformations. They arise naturally in mapping problems involving the function arctan z. Often times they can be employed to map a disk one-to-one and onto a half plane. Let a, b, c, d denote four complex constants with the restriction that . Then the function , is called a bilinear transformation or Mobius transformation or linear fractional transformation.

Theorem 10.3. (The Implicit Formula) There exists a unique bilinear transformation that maps three distinct points onto three distinct points , respectively. An implicit formula for the mapping is given by the equation: .

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(c) John Mathews, 1998, 2006