## Graphing Complex Functions

June 15, 2009 by richbeveridge

In graphing real valued functions, each *x* value chosen is a real number, and each corresponding *y* value is also a real number.

Because both the *x* and *y* values are one-dimensional real numbers, the relationship can be graphed on a plane, showing the *x* and *y* values together only requires TWO dimensions.

We can use the graph of the function relationship between *x* and *y* values to solve equations.

To solve the quadratic equation

0= x²+5x+3

we can graph the function

y=x²+5x+3

and look to see what value(s) of *x* make the *y* be zero.

When we graph the function we look for the *x* values where the graph crosses the *x*-axis. This is because the value of *y* is zero along the *x*-axis.

So the values of *x* that make *y* be zero in the graph of (y=x²+5x+3) are approximately *x*≈-0.70 and *x*≈-4.31

Frequently, we see quadratic (parabolic) graphs that don’t intersect the *x*-axis at all:

This is because there are no real values for *x* that make *y* be zero. In the case of the graph above (y=x²+5x+9) all the *x* values we see along the *x-*axis make *y* a positive number.

Does this mean that there aren’t **any ***x* values that make *y* be zero?

No. If we use the quadratic formula, we can find that there are complex-valued roots that are solutions of the equation

0=x²+5x+9

In this case, (*x*≈-2.5±1.658*i*) are the complex values of *x* that make *y* zero.

If we could see *x* values on the Complex Plane, then we would see these roots on the graph, but, as I mentioned in a previous post, this creates some difficulties.

The picture from Wikipedia that I posted recently is a graph of a different function from the ones above.

Even though it’s not the same graph, it does show one method to try to get around the difficulties of graphing relationships in the Complex Plane.

The picture

shows the Complex Plane of all *x* values. The *y* values are interpreted by the color and intensity of each *x* value on the Complex Plane. The roots (and asymptotes) for the function in this picture are indicated by the points or holes or peaks (however you want to think of them) that we see in the picture.

I haven’t done enough research with this method of graphing to know all the details of how it is colored and how to tell the roots from the asymptotes, but I find it both visually and mathematically beautiful.

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*Related*

on February 15, 2011 at 5:08 PM |Graphing Complex Functions (Again) « Where the Arts Meet the Sciences[...] in June of 2009, I wrote about the issues involved in graphing complex functions. Since then, I’ve been showing this material to my students and discussing the relationship [...]