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# Examples of calculations with complex arithmetic

It is worth running through some simple exercises in order to become familiar with the REC complex number graphing calculator. In fact, the only visible results from the graphing calculator are those shown in the graph rectangle, so the first order of business is to relate the dimensions of this rectangle to the size of the complex numbers, which can be done by placing colored dots at specified locations.

Rarely will just placing complex numbers on the graph plane be of interest. It is more likely that functions will be evaluated, which implies a whole series of argument points and their corresponding function values. But graphing that combination would imply four dimensions, two each for the argument and for the value, and there are only two dimensions available to work with.

One common method of representing a complex function is to choose arguments along the arc of some curve, maybe even a segment of a line parallel to either the real or the imaginary axis. The function values will sit along some other curve, which can be left as it is, or marked somehow with the values of the argument -- say, tick marks at regular argument intervals. Carried further, whole families of arcs, such as coordinate patches, could be mapped into new families, the nature of the function to be deduced from the distortion induced by this mapping. Such factors as the spacing between coordinate lines, or the change of size and orientation of coordinate rectangles would be used to interpret the function.

An alternative is to draw a contour map, in which the arcs drawn represent curves of constant absolute value or of constant phase, or both. Of course, the contours could identify constant real and imaginary parts, but modular contouring seems to be more informative.

Working with absolute value alone, a three-dimensional surface can be imagined as being drawn in perspective. By engraving lines of constant phase upon it, the full function can be displayed. At one time it was fashionable to place a collection of such plaster models in an exhibit case to show that mathematicians actually worked with real objects. Some idea of their beauty can be gleaned from an examination of Jahnke and Emde's classical Tables of Functions with Formulae and Curves [11] with its numerous line drawings of complex functions,

Yet another embellishment is to use stereoviews, which can be either of the three-dimensional absolute value surface seen from the side, or the contour lines seen from directly above. Both are special cases of projections taken from arbitrary angles, and there is no reason that two or more views cannot be combined into the same picture. For example, a perspective view seen to be floating over a contour map sitting just below it.

Next: representation by line images Up: REC/C for Complex Arithmetic Previous: View methods