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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