Meet the Bessel Functions 

>	plot({BesselJ(0,x),BesselY(0,x)},x=0..20,y=-1.5..1.5,title="J_0(x) & Y_0(x)",numpoints=500);

 

Notice that (0)=1, but (x) is unbounded as x-->0. 

Maple has a nice feature to find the roots: 

>	BesselJZeros(0,1);The first zero of (x)

 

>	evalf(BesselJZeros(0,1));The numerical value of the first zero of (x)

 

Look at the plot of above to see this is reasonable. I don't like typing BesselJZeros!   Maple has a nice alias feature: 

>	alias(Z=BesselJZeros):

 

>	seq(evalf(Z(0,n)),n=1..5); The first five positive  zeroes of J_0

 

>	J:=BesselJ(0,r);Jr:=diff(J,r);Jrr:=diff(J,r$2);

 

>	Jrr+Jr/r+J; Clearly J saitisfies the Bessel's DE.

 

Orthogonality 

>	m:=1;n:=2;

 

>	R_m:=BesselJ(0,Z(0,m)*r);R_n:=BesselJ(0,Z(0,n)*r);

 

>	plot({R_m,R_n},r=0..1,title="Two Eigenfunctions");

 

Now  remember the weight function p(r)=r !! 

>	plot(R_m*R_n*r,r=0..1); I_mn:=Int(R_m*R_n*r,r=0..1)=evalf(int(R_m*R_n*r,r=0..1));

 

What happens if you forget the weight function? 

>	plot(R_m*R_n,r=0..1); I_mn:=Int(R_m*R_n,r=0..1)=evalf(int(R_m*R_n,r=0..1));

 

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