Edexcel IAL - Further Pure Mathematics 2- 3.4 Transformations from the z-plane to the w-plane- Study notes  - New syllabus

Elementary Transformations from the \( z \)-plane to the \( w \)-plane

In complex numbers, a transformation maps points from the \( z \)-plane to the \( w \)-plane using an equation of the form

\( w = f(z) \)

Each complex number \( z \) is sent to a new complex number \( w \), giving a transformation of the Argand diagram.

The Transformation \( w = z^2 \)

Let \( z = re^{i\theta} \). Then

\( w = z^2 = r^2 e^{i2\theta} \)

This means:

• The modulus is squared: \( |w| = r^2 \).
• The argument is doubled: \( \arg w = 2\theta \).

So \( w = z^2 \) stretches distances and doubles angles from the origin.

 Linear Fractional Transformations

A transformation of the form

\( w = \dfrac{az + b}{cz + d} \), where \( a,b,c,d \in \mathbb{C} \) and \( ad – bc \ne 0 \)

is called a linear fractional transformation.

It maps lines and circles in the \( z \)-plane to lines or circles in the \( w \)-plane.

Special Cases

• If \( c = 0 \), then \( w = \dfrac{a}{d}z + \dfrac{b}{d} \), which is a combination of rotation, enlargement and translation.
• If \( b = 0 \) and \( c \ne 0 \), the transformation includes inversion.

Geometric Effects

• Multiplying by \( a \) causes a rotation and stretch.
• Adding \( b \) translates the plane.
• Dividing by \( cz + d \) introduces inversion and bending of shapes.