Computer Algebra Recipes

132 CHAPTER 3. LINEAR ODE MODELS
is a linear combination of a Bessel function of the ¯rst kind (BesselJ(m; x)) and
of the second kind (BesselY(m; x)), with C1 and C2 arbitrary constants. In
traditional mathematical notation, the Bessel functions are written as Jm(x)
and Ym(x), m being referred to as the order of the Bessel function. Highlighting
BesselJ(m; x) orBesselY(m; x) on the computer screen will open a Help page
with some (but not much) information about these special functions.
If you wish to know what other special functions are known to Maple, execute
the following inifcns (initially known mathematical functions) command
line and use the hyperlinks provided in the lengthy list of functions. Remember
to close the Help page when you're done.
> ?inifcns;
The Bessel functions are actually in¯nite Frobenius power series solutions of
the Bessel ODE, i.e., a series expansion about x = 0 is sought of the form
y(x) = P1 m=0 cm xm+s . The allowed values of the index s and the forms of the
coe±cients cm are determined5 by substituting y(x) intotheBesselODE.
The series command can be used to obtain the form of the Frobenius series.
For example, since the most commonly occurring order in physical problems
involves positive-integer values of m, Jennifer calculates the Frobenius series of
Jm(x) form = 0 to 2, terms of order x8 and higher being omitted here.
> seq(J[m]=series(BesselJ(m,x),x=0,8),m=0..2);
J0 =1¡ 1
4 x2 + 1
64 x4 ¡ 1
2304 x6 +O(x8 );
J1 = 1 1
x ¡
2 16 x3 + 1
384 x5 ¡ 1
18432 x7 +O(x8 );
J2 = 1
8 x2 ¡ 1
96 x4 + 1
3072 x6 +O(x8 )
Note that at x =0,wehaveJ0 =1andJ1 = J2 =0. Asiseasilycon¯rmedby
increasing the range of m in the seq command, Jm(0) = 0 for m =3; 4 :::
Looking at the above truncated series does not convey much of an idea of
the shapes of J0(x), J1(x), and J2(x). Jennifer will now plot these functions
and attach appropriate identifying labels to each curve. First, in the graph gr1
she plots Jm(x) overtherangex =0to20form =0to2.
> gr1:=plot(fseq(BesselJ(m,x),m=0..2)g,x=0..20,thickness=2):
She uses the textplot command in gr2 to place appropriate labels on the
Bessel function curves. The horizontal and vertical coordinates of the names
(entered as strings) were determined by observing the picture generated by gr1.
> gr2:=textplot([[1.5,0.9,"J0"],[3,0.6,"J1"],[5,0.4,"J2"]]):
The two graphs are superimposed to yield Figure 3.6.
> display(fgr1,gr2g,tickmarks=[2,2]);
The Jm(x) are oscillatory with amplitudes that decrease with increasing x.
Unlike sin(x) orcos(x), the Jm(x) do not cross the horizontal axis at equal
5 A recipe is given in Computer Algebra Recipes for Mathematical Physics [Enn05].