Bessel functions

Mathematica can sum certain power series, such as

Sum[x^k/k!,{k,0,Infinity}]

Sum[(-1)^k/(2k)!x^(2k),{k,0,Infinity}]

Sum[(-1)^k/4^k/(k!)^2 x^(2k),{k,0,Infinity}]

This last one is given as a named function.  This function is called Bessel's function of the first kind of order 0.

Plot[BesselJ[0,z],{z,0,30}]

Plot[BesselY[0,z],{z,0,30}]

Our eigenfunctions are this Bessel function scaled so that a zero of the function lands on the edge of our disk.  We'll illustrate this for a radius 1 disk.

FindRoot[BesselJ[0,z]==0,{z,2}]

α[1]=z/.%

Plot[BesselJ[0,α[1]r],{r,0,1}]

FindRoot[BesselJ[0,z],{z,5}]

α[2]=z/.%

Plot[BesselJ[0,α[2]r],{r,0,1}]

Mathematica has the zeros we need as built-in objects:

BesselJZero[0,34]

α[n_]=BesselJZero[0,n]

Plot[BesselJ[0,α[5] r],{r,0,1}]

Distinct eigenfunctions are orthogonal with respect to an inner product that includes the weight function r:

Integrate[BesselJ[0,α[8]r]*BesselJ[0,α[3]r]*r,{r,0,1}]

α[2]/α[1]

N[%]