Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire

Author:John Derbyshire [Derbyshire, John]
Language: eng
Format: mobi, pdf, epub
Tags: Explore Science. Science : Past and Future : Biographies
Publisher: National Academies Press
Published: 2003-04-15T04:00:00+00:00

FIGURE 13-3 The Riemann surface corresponding to the function z2.

VI. I am going to take a much simpler approach to illustrating complex functions. I'd like you to meet my pal the argument ant, shown in Figure 13-4.

FIGURE 13-4 The argument ant.

The argument ant is awfully hard to see, because he is infinitesimal in size. If you could see him, however, he would look just like a regular ant—a Camponotus japonicus worker, to be precise—with the regulation number of appendages, antennae, etc. In one of the frontmost appendages, which for convenience we may call a “hand,” the argument ant holds a small gadget rather like a beeper, or a mobile telephone, or one of those global positioning devices that can tell you exactly where you are. This gadget (Figure 13-5) has three displays. The first display, labeled “Function,” shows the name of some function: z2, log z, or whatever the gadget might be set to. The second display, labeled “Argument,” shows the point—the complex number—the argument ant is currently standing on. The third display, labeled “Function value,” shows the value of the function at that argument. So the argument ant always knows exactly where he is; and, for any given function, he knows where the point he's standing on gets sent to by the function.

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