Bessel functions are sometimes called cylindrical functions because they arise naturally from physical problems stated in cylindrical coordinates. Bessel functions have a long list of special properties that make them convenient to use. But because so much is known about them, introductions to Bessel functions can be intimidating. Where do you start? My suggestion is to start with a few properties that give you an idea what their graphs look like.

A textbook introduction would define the Bessel functions and carefully develop their properties, eventually getting to the asymptotic properties that are most helpful in visualizing the functions. Maybe it would be better to learn how to visualize them first, then go back and define the functions and show they look as promised. This way you can have a picture in your head to hold onto as you go over definitions and theorems.

I’ll list a few results that describe how Bessel functions behave for large and small arguments. By putting these two extremes together, we can have a fairly good idea what the graphs of the functions look like.

There are two kinds of Bessel functions, J(z) and Y(z). Actually, there are more, but we’ll just look at these two. These functions have a parameter ν (Greek nu) written as a subscript. The functions J ν (z) are called “Bessel functions of the first kind” and the functions Y ν are called … wait for it … “Bessel functions of the second kind.” (Classical mathematics is like classical music as far as personal names followed by ordinal numbers: Beethoven’s fifth symphony, Bessel functions of the first kind, etc.)

Roughly speaking, you can think of J‘s and Y‘s like cosines and sines. In fact, for large values of z, J(z) is very nearly a cosine and Y(z) is very nearly a sine. However, both are shifted by a phase φ that depends on ν and are dampened by 1 /√z . That is, for large values of z, J(z) is roughly proportional to cos(z – φ)/√z and Y(z) is roughly proportional to sin(z – φ)/√z.

More precisely, as z goes to infinity, we have

and

These statements hold as long as |arg(z)| < π. The error in each is on the order O(1/|z|).

Now lets look at how the functions behave as z goes to zero. For small z, J ν behaves like zν and Y ν behaves like z-ν. Specifically,

and for ν > 0,

If ν = 0, we have

Now let’s use the facts above to visualize a couple plots.

First we plot J 1 and J 5 . For large values of z, we expect J 1 (z) to behave like cos(z – φ) / √z where φ = 3π/4 and we expect J 5 (z) to behave like cos(z – φ) / √z where φ = 11π/4. The two phases differ by 2π and so the two functions should be nearly in phase.

For small z, we expect J 1 (z) to be roughly proportional z and so its graph comes into the origin at an angle. We expect J 5 (z) to be roughly proportional to z5 and so its graph should be flat near the origin.

The graph below shows that the function graphs look like we might have imagined from the reasoning above. Notice that the amplitude of the oscillations is decreasing like 1/√z.

Next we plot J 1 and Y 1 . For large arguments we would expect these functions to be a quarter period out of phase, just like cosine and sine, since asymptotically J 1 is proportional to cos(z – 3π/4) / √z and Y 1 is proportional to sin(z – 3π/4) / √z. For small arguments, J 1 (z) is roughly proportional to z and Y 1 (z) is roughly proportional to -1/z. The graph below looks as we might expect.

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