Introduction

Transformations and Linear Mappings

In the study of complex analysis the elementary transformation of the form is called a linear transformation and it is a one-to-one mapping of the complex z-plane onto the complex w-plane. Any geometric object is mapped onto an object that is similar to the original object: hence linear transformations can be called similarity mappings. Notice that the usage of the word "linear" is different than that used in linear algebra.

Definition 1. The transformation

is a one-to-one mapping of the z-plane onto the w-plane and is called a translation.

Definition 2. Let be a fixed real number. Then the transformation

is a one-to-one mapping of the z-plane onto the w-plane and is called a rotation.

Definition 3. Let K > 0 be a fixed positive real number. Then the transformation

is a one-to-one mapping of the z-plane onto the w-plane and is called a magnification.

Definition 4. Let and where K > 0 is a positive real number. Then the transformation

is a one-to-one mapping of the z-plane onto the w-plane and is called a linear transformation. It can be considered as the composition of a rotation, a magnification, and a translation.

It is easy to see that translations and rotations preserve angles. Since magnifications rescale distance by the factor K, it follows that triangles are mapped onto similar triangles, and so angles are preserved. Since a linear transformation can be considered as a composition of a rotation, a magnification, and a translation, it follows that a linear transformations preserve angles. Consequently, any geometric object is mapped onto an object that is similar to the original object: hence linear transformations can be called similarity mappings.

Basic Properties of Conformal Mappings

A mapping w = f(z) is said to be angle preserving, or conformal at , if it preserves angles between oriented curves in magnitude as well as in orientation. It is useful to be able to graph complex functions and observe that if f(z) is analytic and then the mapping is conformal and orthogonal curves are mapped onto orthogonal curves.

Theorem 9.1. Let f(z) be an analytic function in the domain D, and let be a point in D. If , then f(z) is conformal at .
Proof.

Theorem 9.2. Let f(z) be analytic at . If
and , then the mapping magnifies angles at the vertex by the factor k.
Proof.

Bilinear 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 9.3. (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: .
Proof.

Return to the Complex Analysis Project

(c) John Mathews, 1998, 2006