Mathematics 373 – Applied Mathematics (PDE) Lab Day on Bessel Functions April 19, 2001

Background

Last time, we introduced the Bessel functions of the first kind via the series expansions:

(1)  J_{m}(z) = ∑_{ℓ=0}^{∞} \frac{(−1)^{ℓ}}{ℓ! (m + ℓ) !} (\frac{z}{2})^{2ℓ+m}

Using the Ratio Test, we saw that the series for J_{m}(z) in (1) converges for all z ∈ R, so it gives a well-defined function on R. (Actually, it converges for complex z as well, but that is another story!) At this point in your acquaintance with them, though, Bessel functions probably seem like “magic” since apart from the name J_{m}(z) we have only the description (1). Today, we want to get more familiar with these functions and their properties using Maple to generate plots, work with some of the formulas, etc.

Maple Details

The Bessel functions J_{m}(z) are built-in in Maple. The Maple notation for this function is BesselJ(m,z). You can think of BesselJ(m,z) as a function of two variables – the first (namely the m) is the order, the second is the actual argument, z. Since these functions are built-in, you can use them in a Maple session just like the elementary functions sin, cos, tan, exp, ln, arctan and so forth. For instance you can operate on Bessel functions with other Maple commands, you can plot them or functions made using them and other functions, etc.

Lab Questions, part 1

A) You can recover the series expansion (1) for J_{m}(z) in the case that m is a positive integer using the taylor command in Maple. Try this command:

                       taylor(BesselJ(2,z),z=0,11);

The output is the terms of degree < 11 in the Taylor series of J_{2}(x) at x = 0. Does this agree with (1) above? Check using Maple’s sum command to write out the same terms as in (1) with m = 2:

              sum((-1)^l/(l!*(2+l)!)*(z/2)^(2*l+2),l=0..10);

B) Plot J_{0}(z), J_{1}(z), J_{2}(z), J_{3}(z) together on the same set of axes, with z ∈ [0, 10]. Which function is which and how can you tell? Note: In (1), the term of lowest degree (in z) in the series for J_{m}(z) is z^{m}. What does that tell you about the shape of the graph y = J_{m}(z) for z near zero?

C) The Bessel functions and their derivatives satisfy a plethora of different relationships. For example, try entering the command

                           diff(BesselJ(m,z),z);

to compute J′_{m}(z). Note that the result is expressed in terms of other Bessel functions. How does this formula relate to the formulas given on page 512 of the text? Next, what happens if you compute the second derivative, J′′_{m}(z) using

                          diff(BesselJ(m,z),z$2);

How does the result here compare with the relation between J_{m}(z), J′_{m}(z), J′′_{m}(z) given by Bessel’s equation of order n? (It may help to simplify your results.)

More Maple Details

Note that the graph y = J_{m}(z) for m ≥ 0 always looks more or less like a decaying oscillation. For us, the important consequence is that as z → ∞, we get a sequence of positive roots of the equation J_{m}(z) = 0 (a different sequence for each m). These roots of the Bessel functions will be very important in our study of the vibrating circular drumhead. In addition to the BesselJ function, Maple also has a function called BesselJZeros. Try entering

                         evalf(BesselJZeros(3,4));

The result is a decimal approximation to the 4th positive zero of J_{3}(z). (Note: “positive” means strictly positive here.) Compare with the plot of y = J_{3}(z) to check this. If you leave out the evalf in the command above, Maple will just give you the input back again, since that is an exact way to specify the number in question (!) In general BesselJZeros(m,n); gives the nth (strictly) positive root of J_{m}(z) = 0.

Lab Questions, part 2

D) (“The punchline” – be sure you understand what is happening here and in the following questions E,F carefully.) We introduced the Bessel equation of order m:

z^{2} \frac{d^{2}f}{dz^{2}} + z \frac{df}{dz} + (z^{2} − m^{2})f = 0

because we wanted to study the eigenvalues and eigenfunctions for the boundary value problem:

(2)  \{ \begin{matrix}\frac{d}{dr} (r \frac{df}{dr} ) − \frac{m^{2}}{r} f = −λrf \\ f(a) = 0 \\ |f(0)| < ∞ \end{matrix} .

(We got Bessel’s equation by letting z^{2} = λr^{2}.) Let

L(f) = \frac{d}{dr} (r \frac{df}{dr} ) − \frac{m^{2}}{r} f

denote the differential operator in (2). Using the Bessel function zeroes, we can construct the functions

              f := BesselJ(m,BesselJZeros(m,n)*r/a);

Compute L(f) using Maple, and simplify. What does the result tell you?

E) Take m = 4, n = 1, 2, 3, 4, 5 (one at a time of course), a = 2 and plot the functions f from part F on the interval r ∈ [0, 2]. Do these functions satisfy the properties we know for eigenfunctions of Sturm-Liouville boundary value problems? Explain.

F) What are the eigenvalues and eigenfunctions of the general problem (2)? Give a complete description by generalizing what you did in parts D and E.

Assignment

Group write-ups for the above lab questions, due April 26.