Can All Functions Be Plotted on a Complex Plane?

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SUMMARY

All functions can be plotted on a complex plane, but the representation differs from the standard xy-plane due to the unique definition of the dot product. While recursive sequences like the Mandelbrot Set are commonly visualized, one can also plot x,y equations by ignoring the real part of the y output. The complex plane allows for the visualization of fractals, utilizing the properties of imaginary numbers in ways that the xy-plane cannot achieve.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with fractals, specifically the Mandelbrot Set
  • Knowledge of plotting functions in mathematical software
  • Basic concepts of dot products in vector spaces
NEXT STEPS
  • Explore the properties of complex numbers in depth
  • Learn how to visualize the Mandelbrot Set using software like MATLAB or Python
  • Research the mathematical definitions of dot products in complex vector spaces
  • Investigate other types of fractals and their representations on the complex plane
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Mathematicians, educators, students in advanced mathematics, and anyone interested in visualizing complex functions and fractals.

scott_alexsk
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complex plane

Hello, I was wondering if there are only specific types of forumlas that you can graph on a complex plane. I mean can you only plot recursive sequences such as the Mandelbrot Set or can you also plot x,y equations while just ignoring the real part of the y output.
Thanks,
-scott
 
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Well, to start off, the "only" difference between the complex plane and the standard xy-plane is how the dot product is defined. Now how the plot functions can be used depending of the software is a vague question.
Fractals, as you mentionned, use the imaginary part of the number(and its properties) in a way that no xy-plane can, even if it can plot recursive functions.
 

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