Equations break down at certain special places. Sneaking up on those critical locations from various directions can be most enlightening. Think about the simple reciprocal function, "1/x" — what happens when x = 0? Yes, it "blows up" — but how? Come at zero from the positive side and 1/x gets bigger and bigger; approach from negative numbers and 1/x heads toward minus infinity. Move toward zero via other routes in the complex plane and find other values.

Understanding the singularities of certain functions tells one everything there is to know about them. It's like figuring the shape of a drum based on the sounds that come out when you strike it, or deducing the configuration of electrical charges inside a black box by measuring the fields on the surface. Some quantities are tightly interwoven with each other: change one of them, and the others must follow. Singularities are often like that — anchors that hold everything else in place.

Tuesday, February 01, 2000 at 05:48:34 (EST) = 2000-02-01


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