Big on the edges

I finally got my grades today. Strangely enough, I got an A- for complex analysis. To the best of my knowledge, I suck in this subject (to the point that I think I never passed an exam.) Mirabile dictu, or something. The best that I can recall from my time spent on that subject (aside from an endless influx of drawings of contours) would be this tiny tidbit.

There is a theorem in complex analysis called the maximum modulus principle. Simply put, it says that a complex valued function analytic in a domain will have no maximum value within that domain. A consequence of this is that all maximum values of the function (if they exist) will occur on the contour enclosing the domain itself.

What does all that babble mean? Well, it just says that if I have a closed space and my function has values there, there is no area there that has a bigger value than everyone else. It's perfect competition at its finest; at most, the areas would have to be equal to each other (going back to our function, it would have to be constant-valued.) The consequence above follows from the theory almost intuitively (quite dangerous in mathematics, but a little proving exercise should validate this assumption.) That implication, on the other hand, has an application in a field significantly removed from such abstractions as complex analysis; that is, the field of operations research.

Consider a maximization problem. For example: you are assembling three different products using a set of raw materials, and you want to produce a certain number of each product such that you use up all your available material and (assuming you sell all the products you have made) yield the maximum possible profit. A method in operations research called linear programming will lay down the conditions that restrict your problem and offer a solution. One way would be to place all the conditions together in a graph, forming a polygon. Now, applying the maximum modulus principle, the maximum value for our function (in this case, the profit) would have to lie not inside the region but on the edges of the polygon itself.

Amazing stuff, eh? A simple solution to a practical problem, backed by an abstruse amount of abstraction. I guess therein lies the beauty of mathematics; to break things down to the finest, most incomprehensible core and reconstruct it into an elegant framework that makes things, if not easier, bearable.