Computer Algebra Recipes

3.3. SPECIAL FUNCTION MODELS 131
3.3.1 Jennifer Introduces a Special Family
\That's a great deal to make one word mean," Alice said in a
thoughtful tone. \When I make a word do a lot of work like that,"
said Humpty Dumpty, \I always pay it extra."
Lewis Carroll, Through the Looking Glass, 1872
Although we shall not emulate Humpty Dumpty and pay the word \special"
extra for making it work so hard in this section, we would tell Alice, if she were
here, that the phrase \special" function does encompass a great many functions
with their associated mathematical properties. The Handbook of Mathematical
Functions by Abramowitz and Stegun [AS72], for example, contains over
1000 pages dealing with the properties of special functions. This handbook,
produced by the U.S. National Bureau of Standards, is one of the standard
reference books for these functions and has been reprinted many times since it
was ¯rst issued.
We have asked our MIT mathematician friend Jennifer to introduce two of
the more prominent members of this special family of functions.
Letting the letter L stand for ¸, Jennifer forms a functional operator SL for
generating speci¯c cases of the Sturm{Liouville ODE when the forms of p, q,
w, andLare supplied as arguments.
> restart: with(plots):
> SL:=(p,q,w,L)->diff(p*diff(y(x),x),x)-q*y(x)=-L*w*y(x);
μ μ μ
d d
SL := (p; q; w; L) ! p
dx dx y(x)
¡ q y(x) =¡Lwy(x)
A second functional operator sol is introduced to provide the general analytic
solution y(x) of a speci¯ed ODE.
> sol:=ode->dsolve(ode,y(x)):
Now Jennifer will make two choices for p, q, w, andLthat lead to probably
the best-known two members of the family of special functions. Taking p = x,
q = ¡x, w =1=x, andL = ¡m2 as arguments in the Sturm{Liouville operator
produces ode1 .
> ode1:=SL(x,-x,1/x,-m^2);
μ
d
ode1 :=
dx y(x)
μ
2 d
+ x y(x) + x y(x) =
dx2 m2 y(x)
x
To illustrate that Maple is able to identify this ODE, Jennifer loads the DEtools
package and applies the odeadvisor command to ode1 .
> with(DEtools): odeadvisor(ode1);
[ Bessel]
So, ode1 is Bessel's di®erential equation, whose general solution y(x)
> sol(ode1);
y(x) = C1 BesselJ(m; x)+ C2 BesselY(m; x)