Equation of Line in Complex Plane

Theorem

Formulation 1

Let $\C$ be the complex plane.

Let $L$ be a straight line in $\C$.

Then $L$ may be written as:

$\beta z + \overline \beta \overline z + \gamma = 0$

where $\gamma$ is real and $\beta$ may be complex.

Formulation 2

Let $\C$ be the complex plane.

Let $L$ be the infinite straight line in $\C$ which is the locus of the equation:

$l x + m y = 1$

Then $L$ may be written as:

$\map \Re {a z} = 1$

where $a$ is the point in $\C$ defined as:

$a = l - i m$