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Complex components of simple functions

 Parabola

This is the graph of y = x2.
The curve does not fully occupy the graph. What values of x result in negative values of y? The answer is that y is negative when x has 'imaginary' values (ix = x  times the square root of -1). Add an 'imaginary' axis to the graph. This part of the curve (coloured red below) occupies part of the complex ix, y plane for values of y corresponding to imaginary values of x.
Note that each value of y, from -infinity to +infinity, has two values of x (two roots).

This type of equation requires an imaginary axis to fully display its graph.

Cubic

Let's look at the graph of  y = x3.
The curve does  fully occupy the graph but whereas the graph of y = x2 has two values of x for each value of y, the graph of  y = x3 should have three. The graph shows only one, so where are the other two?  The answer is that they are in the complex x, ix,y space. This part of the curve is coloured red in the following graph.
This type of equation requires an imaginary axis to fully display its graph.

Circle

Here is a graph of a function where neither the y nor the x axis appears fully occupied.

x2 + y2 = r2 is the equation that describes a circle, radius r, centred on the origin.

Rewrite the equation as y = (r2 - x2)1/2 and calculate and plot values of y for (- r) <= x <= r. This gives a graph of a circle in the real x,y plane.

When x <=(- r) and x >= r the graph is a rectangular hyperbola in the complex x,iy plane.

The structure expanded

Another way to explore a graph is to add a constant to the right hand side of the equation. This has the effect of moving the curve up or down the y axis. A curve can then be plotted from the values of the roots of the equation for each value of k.

Alternatively, for this particular equation, you can rewrite the equation as x = (r2 - y2)1/2 and calculate and plot values of x for y <= (- r) and y >= r.

A third way is to plot values of y for various imaginary values of x. (ix)

If you do any of these,  you find that the curve has another complex component when y <= (- r) or y >= r. It is another rectangular hyperbola, this time in the complex ix,y plane. 

Is there more than one imaginary direction?

The next question is “Is the ix direction the same as the iy direction?”

The answer can be found by changing the sign of the terms of the original equation. Changing the sign of a term rotates the curve into different planes. (x and y represent positive real numbers)

Consider only the circle part of the curve, for real values of x, y and r.
  • x2 + y2 = r2 defines a circle in the real x,y plane.
  • x2 - y2 = r2 defines a circle in the complex x,iy plane.
  • - x2 + y2 = r2 defines a circle in the complex ix,y plane.
  • - x2 - y2 = r2 defines a circle in the imaginary ix,iy plane.
Since a circle can be defined in the ix,iy plane, the ix direction must be different from the iy direction.

So, the curve defined by the equation x2 + y2 = r2 requires four axes, two real and two imaginary.

By a similar argument the graph of the equation for a sphere, x2 + y2 + z2 = r2 can be shown to require six axes, three real and three imaginary. ( x, y and z can be positive, negative, real or imaginary numbers)

In general, it seems that a similar equation with n variables requires n real and n imaginary axes to fully display its graph.

Mike Holden - Sep 2005
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