Sunday Function

If you go to the bank and open a savings account, the banker might tell you about the virtues of compound interest. He may say something like "Even if you never deposit anything, the rate of change of the money in your account is proportional to the amount of money in your account. The more you earn, the faster you earn." Now the banker is less likely to use such explicitly mathematical language, but what he's telling you is actually a differential equation. A differential equation is an equation that relates a quantity to the rate of change of that quantity. The banker is relating your balance to the rate of change of your balance, and without invoking mathematical notation he has given a differential equation nonetheless. Mathematically it's pretty easy to find the solution to that equation, which describes the balance in your account as a function of time. It happens to be P*e^(r*t), where P is your initial deposit, r is the interest rate, and t is the time in years.

Pretty much all of physics is differential equations when you come right down to it. Expressed in words, this Sunday's function is the solution to a more complicated differential equation that crops up quite a bit in various physical contexts. Let me see if I can do it in words, pretending again that the quantity of interest is an account balance:

"Take the amount of time that has gone by since you opened your account, and square it. Multiply that by the rate of change of the rate of change of your balance. Now take the amount of time that has gone by since the opening, and multiply it by the rate of change of your balance. Add both of those results together. Now square the time once again, and subtract the square of a constant number α (you get to pick that number) and multiply that difference by the amount of money in your account. Add that to the sum of your previous two results. Your balance as a function of time will change in such a way as to ensure that the sum of those three parts is always zero."

And that is just too horrible to do anything with, which is why we use mathematical notation. More compactly and elegantly, the wall of text above can be written as:

Which we in the business call Bessel's equation. Believe it or not, it's not too hard to work with. Its solutions are called the Bessel functions.

Unlike the first banker's differential equation for compound interest, you may notice that the Bessel equation doesn't just deal with the rate of change. It also deals with the rate of change of the rate of change. This makes Bessel's equation a second order differential equation. Second order equations have the property that they don't just have one type of solution, but in fact they have two linearly independent solutions. In this case they are uncreatively called the Bessel functions of the first and second kinds.

Here (with α = 0, 1, and 2) are the Bessel Functions of the first kind:

And here (with the same three alphas) are the Bessel functions of the second kind:

The parameters α can be anything including real and complex numbers but most of the time the integers {0, 1, 2... } suffice to solve physics problems.

The biggest difference between the two kinds of functions is that the functions of the second kind blow up at the origin whereas (except for α = 0) the functions of the first kind are zero at the origin.

Often that's a pretty useful thing to know, since solutions that blow up within the domain of your problem are usually excluded on physical grounds. In problems where the origin is not within the domain of your problem (such as hard-sphere scattering in quantum mechanics) in general the solutions of the second kind will still be necessary to solve the problem.

For differential equations of higher order, there will be even more linearly independent solutions. At that point we usually let the mathematicians worry about them.

Oh, and if the banker offers you a Bessel function account, don't take it. All the Bessel functions (of positive α) approach zero as t become large.