The transcomplex surfaces are one of the most dramatic manifestations of the transcomplex number system. It is in this stage that you will see one of the benefits of using a unified plotting system instead of using the traditional two separate complex planes to plot a single complex function. In addition to that, it is here that we will see how the graphs of real functions are particular cases of the graphs of more generalized transcomplex graphs.The reader is invited to compare the plots he/she will see in each one of the functions, and in each one of the screen captures, with the same problems as worked in any complex variable textbook, and any mathematical reference he/she have in hand and reach at his/her own conclusions.
In its real numbers expression, the real quadratic function is written as
y = x2.
Within the complex number system, it is expressed as
W = (x + iz)2.
Since the transcomplex numbers are the complex numbers extended, the notation is similar, but the way that this function is plotted is what makes the big difference.
y = f(x),
The wavy sinusoidal real function within the complex numbers is written as
W = sin(x + iz).
For comparison purposes, there is also a page of quick links about visualization of complex variables. All of them are collected in this page: Complex variables visualization pages .