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(Here’s another tidbit which originally appeared on my physics blog.)

(Warning: Much of this was written in a cloud of decongestants. If there are any errors, well, you know why….)

Infinity can be a slippery concept, and it causes no end of woes to mathematicians. But, over the years, they have gotten a better and better handle on the concept. This was helped greatly by the work of Georg Cantor, who developed the basic mathematical tools used today for grappling with infinities. But he was by no means the first, nor the last. The legendary 18th century mathematician Leonhard Euler made great strides in devising methods for dealing with divergent infinite series, and the definitive work on that subject is the 1949 book by G. H. Hardy,  Divergent Series.

Let’s get a common misconception out of the way: Something divided by zero most certainly does NOT equal infinity. Division by zero is undefined, as the concept is absolutely meaningless by any mathematical definition of division. What is the case is that, in the limit as the divisor of an expression approaches zero, the value of the expression goes to either positive or negative infinity. (Depending upon the function, it can be either positive or negative depending upon which direction the limit is taken from.) A function exhibiting such behavior is said to have a discontinuity at that point.

Knowing how to deal with infinities and divergences can be crucial. The need to tame divergences in the blackbody radiation problem led Planck to take the first step in creating quantum mechanics (although, to be fair, Planck basically reformulated the problem in a form that didn’t result in infinities).  Quantum electrodynamics calculations required the creation the “dippy procedure” of renormalization. And, on the bleeding edge of theoretical physics, the challenge of reconciling quantum theory and general relativity is fraught with seemingly intractable divergences. But such heady problems aren’t the only places where divergences crop up. They can arise in the most seemingly simple math problems. Read More

(This originally appeared over on my physics blog. Enjoy!)

Lately, as a form of review, I’ve been taking a quantum mechanics course on Coursera. (It was, in fact, that course which prompted me to recently post a derivation of the Schrödinger equation a few weeks ago.) A couple of the lectures were devoted to a brief introduction to Feynman’s path-integral formulation of quantum mechanics, something typically not brought up in courses at that level, which was a refreshing change of pace. A key component of deriving Feynman’s approach is Laplace’s method, a mathematical technique that I’ve probably not thought about since taking Mathematical Methods for Physicists way back in the Dark Ages when I rode a dinosaur to grad school. (Now, where the heck is my copy of Arfkin?) A review was definitely in order.
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