Equation of Line in Complex Plane/Formulation 1

Theorem
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.

Proof
From Equation of Straight Line in Plane, the equation for a straight line is:


 * $A x + B y + C = 0$

Thus:

The result follows by setting $\beta := \dfrac A 2 + \dfrac B {2 i}$ and $\gamma := C$.