Some Bessel function properties in Mathematica

(∫_0^∞)BesselJ[1,k r]BesselJ[1,q r]r\[DifferentialD]r

(∫_0^R)BesselJ[0,k r]r\[DifferentialD]r

BesselJZero[1,3]//N

N[BesselJZero[0,1],32]

Demonstration of Bessel orthogonality over a finite range

R=1; k=BesselJZero[0,1]; q=BesselJZero[0,2];

k//N

q//N

(∫_0^R)BesselJ[0,k r]BesselJ[0,k r]r\[DifferentialD]r

(∫_0^R)BesselJ[0,k r]BesselJ[0,q r]r\[DifferentialD]r

Plot[BesselK[1,x],{x,0.1,5}]

Plot[{BesselJ[0,x],BesselY[0,x]},{x,0.2,20}]

Plot[{BesselY[0,x],BesselY[1,x]},{x,0.1,15}]

Plot[{BesselI[0,x],BesselK[0,x]},{x,0.1,3}]

BesselJZero[0,10]//N

Table[BesselJZero[0,k]//N,{k,10}]