Today we have an interesting post about a “silly function”:

Sometimes you’ll be happily calculating along, and wind up with an equation you wouldn’t want to meet at night in a dark alley. But, a quick flip through Abramowitz & Stegun frequently turns up your nemesis, along with handy tricks for disarming it. Additional satisfaction comes when you write the paper, and get to throw off lines like “The solutions to equation 4 are confluent hypergeometric functions (of the first kind)”.

However, it will be hard top my amusement when I once discovered that the solutions to my problem were closely related to Anger functions.

Those who have taught differential equations will note the similarity to the Bessel Function of the first kind. In fact, this is a particular solution to:

Note: there is a whole library of interesting functions here.

**An interesting issue in elementary probability**

Sean Carrol makes a point about non-normalized probability with this delightful little example:

An eccentric benefactor holds two envelopes, and explains to you that they each contain money; one has two times as much cash as the other one. You are encouraged to open one, and you find $4,000 inside. Now your benefactor — who is a bit eccentric, remember — offers you a deal: you can either keep the $4,000, or you can trade for the other envelope. Which do you choose?

If you do the usual “expected value” calculation, then your strategy should not matter no matter how much money you see in the envelope and how big of the factor there is between the two envelopes (e. g., one envelope might contain, say, 1,000 times more money than the other).

But now suppose the factor is that one envelope has, say, 1000 times more money than the other and you open your envelope and see 2 dollars. Of course, you’d switch. On the other hand, if your envelope had 10,000 dollars in it, then of course you wouldn’t. The real life complication is that there is a realistic upper bound in how much could be in the envelope and if you discover that your envelope contains something near that upper bound, then you won’t switch.

In short, the “real life” problem is really a conditional probability problem: “what is the probability that my envelope contains the larger amount given that the upper bound is roughly X dollars”.