plotcompare(f(z), g(z), z = a+c*I..b+d*I, options);

plotcompare(f(z) = g(z), z = a+c*I..b+d*I , options);

plotcompare(f, g, a+c*I..b+d*I, options);

plotcompare(f = g, a+c*I..b+d*I, options);

A comparison of  with its cubic Taylor polynomial. (Note that if there is no imaginary part, as in this example, the graphs of  and  will be empty.)

If you shift the range, for example to the right by 10, the function and its cubic approximation do not look alike anymore.

Is the cube root of -8 equal to -2? In Maple, the function  means the principal root, which is not real for .  Hence the following are not the same for negative real x.

Comparing the same expressions but using the same_box option and optional arguments of plot for thickness and coloring:  appears in blue and and  in green.

In the above plots, for positive values of  both the real and imaginary parts of the expressions being compared are the same while for negative values of  they are different. A 3-D plot can make the regions where these expressions are equal or different more evident. (Try rotating this plot with the mouse and using the same_box option).

The following example shows two expressions which differ in value only over a line, so that the 3-D plots are visually the same.

Though not visible, the difference is in the value of the imaginary parts,  and , when taken over real and positive values of . Note the branch cut. To examine the cut, you can rotate the plots above with the mouse.

The procedure recommended to turn the difference (if any) visible in cases like this, that is, whenever there are branch cuts in the 3-D plots, is to use plotcompare to generate 2-D plots, for instance giving a real range or using the assuming facility together with the optional argument same_box.

In above, these expressions are not valid for real (Im(x) = 0) and negative values of x. For any other point in the complex plane not in RealRange(-infinity, 0) these expressions have the same value as suggested by the 3-D plots. For instance, the following two pairs of plots the for purely imaginary values of  are equal.

In the two input lines above you can also use the option scale_range = N together with assuming, so that the range for the plots will respectively be x = -N..N and x = -N*I..N*I.

The following suggests that

while this other plot shows that :

This is the plot of the "expression" (in this case a function) BesselK(1/2,z), illustrating the branch cut of BesselK for Re(z) < 0

The branch cut is visible in the second plot box for the imaginary part of the function and is located on the real axis, negative side. To manipulate each of the plots further enter  for the real box plot or  for the imaginary box plot, like in:

The following example illustrates the use of a procedure to input the expressions being plotted. The real and imaginary parts of  and  are equal only for .

To see where  differs from  input:

By constructing a piecewise function , the plots become the same. This uses the same_box option to superimpose the plots for  and .

The default range of the plots below is -1-I..1+I.

Consider the following identity relating the Airy and Bessel functions, taken from the Handbook of Mathematical Functions by Abramowitz and Stegun. The following plots show that this identity is, in fact, not valid over the whole complex plane.  This is the command to create the plot from the Plotting Guide.

The region of validity can be explored visually by superimposing the plots of the real parts and then the imaginary parts of the two functions. You can do this using the same_box option or without recomputing the plots taking advantage of the  variable to which the array of plots was assigned when calling plotcompare.

The region of validity of the identity can also be explored by plotting for z real or imaginary.

Consider the following identity.

The Maple simplify command is not able to prove that this identity is valid for arbitrary values of a.

Although you cannot prove this identity using plotcompare, it does provide a quick indication that the identity is true for different values of the parameter .

This identity can in fact be verified exactly for arbitrary values of , if you first convert to hypergeom.

The command to create the 2-D plot from the Plotting Guide is

convert/to_special_function, plot, plot3d, plots, plots[display], RealRange