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APPENDIX A A MAPLE Primer
A.6
A.7
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Special Functions
In MAPLE, J n (x) is called BesselJ(n,x), and the Bessel function of the second kind, Y n (x),
is denoted BesselY(n,x). To evaluate, for example, J 3 (1.21),use
evalf(BesselJ(3,1.21));
In similar fashion, there are BesselI and BesselK commands for modified Bessel
functions of the first and second kinds, respectively.
For integrals involving Bessel functions, use the int command. For example,
∗
∗
∗
evalf(int(x BesselJ(1,x) cos(3 x),x=0..1));
1
will give a decimal evaluation of xJ 1 (x)cos(x)dx.
0
We can find (approximately) the kth (in order of increasing magnitude) positive zero of
J n (x) by using
evalf(BesselJZeros(n,k));
For example,
evalf(BesselJZeros(3,7));
returns the seventh positive zero of J 3 (x).
The nth Legendre polynomial P n (x) is denoted LegendreP(n,x) in MAPLE. Since
Legendre’s differential equation is second order, there is a second, linearly independent solu-
tion, often denoted Q n (x). In MAPLE, this function is LegendreQ(n,x). To obtain P n (x)
explicitly, use
simplify(LegendreP(n,x));
Integrals involving Legendre polynomials can be done using the int command. For example,
∗
evalf(int(sin(x) LegendreP(5,x),x=-1..1));
1
will give a decimal evaluation of sin(x)P 5 (x)dx.
−1
Legendre polynomials form a special case of a class of special functions called orthogonal
polynomials, which include Laguerre polynomials, Hermite polynomials and many others. The
command
with(orthopoly);
will call up MAPLE’s subroutines of orthogonal polynomials.
Complex Functions
MAPLE will do complex arithmetic with the imaginary unit i denoted I.
For the residue of a function at a point, use the command
Res( f, z 0 );
For example, if f (z) = cos(z)/z, entered by
f:=z → cos(z)/z;
then the residue at zero can be computed as
residue(f(z),z=0);
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