P[r_,x_] = (1-r^2)/(2 Pi(1+r^2-2r Cos[x]));

Plot[{P[0.3,x],P[0.5,x],P[0.7,x],P[0.9,x]},{x,-Pi,Pi},PlotRange->{0,1},PlotStyle->Thick,Filling->Axis]

Plot[{BesselJ[0,x],BesselJ[1,x],BesselJ[2,x]},{x,0,15},PlotStyle->Thick,Filling->Axis]

BesselAsy[n_,x_]=Sqrt[2/(Pi x)] Cos[x-Pi/4 - Pi n/2];

BesselPower[n_,x_,k_]=Sum[(-1)^q/(q!(n+q)!)(x/2)^(n+2q),{q,0,k}];

Plot[{BesselPower[0,x,5],BesselPower[0,x,4],BesselAsy[0,x],BesselJ[0,x]},{x,0,8},PlotRange->{-.5,1.2},PlotStyle->Thick,PlotLabel->(Subscript[J, 0])]

Plot[{BesselPower[1,x,5],BesselPower[1,x,4],BesselAsy[1,x],BesselJ[1,x]},{x,0,8},PlotRange->{-.5,1},PlotStyle->Thick,PlotLabel->(Subscript[J, 1])]

Plot[{BesselPower[2,x,5],BesselPower[2,x,4],BesselAsy[2,x],BesselJ[2,x]},{x,0,15},PlotRange->{-.5,1},PlotStyle->Thick,PlotLabel->(Subscript[J, 2])]

Plot[{BesselPower[3,x,5],BesselPower[3,x,4],BesselAsy[3,x],BesselJ[3,x]},{x,0,15},PlotRange->{-.55,1},PlotStyle->Thick,PlotLabel->(Subscript[J, 3])]

LogPlot[{BesselJ[0,x],BesselJ[1,x]},{x,500,1000}]

z=Array[BesselJZero,{5,5},{0,1}]//N;

z//TableForm

z[[1]][[1]]

en=DensityPlot[BesselJ[0,r*z[[1]][[2]]],{r,0,1},{theta,-Pi,Pi},ColorFunction->(ColorData["Rainbow"][#]&),PlotPoints->36];

Show[en/.GraphicsComplex[pnts_,data__]:>GraphicsComplex[#[[1]]*{Cos[#[[2]]],Sin[#[[2]]]}&/@pnts,data],PlotRange->All]

Show[DensityPlot[BesselJ[1,r*z[[2]][[2]]]*Cos[2*theta],{r,0,1},{theta,-Pi,Pi},ColorFunction->"DeepSeaColors",PlotPoints->36]/.GraphicsComplex[pnts_,data__]:>GraphicsComplex[#[[1]]*{Cos[#[[2]]],Sin[#[[2]]]}&/@pnts,data],PlotRange->All]

Grid[Table[Show[DensityPlot[BesselJ[j,r*z[[j+1]][[k+1]]]*Cos[j*theta],{r,0,1},{theta,-Pi,Pi},ColorFunction->"DeepSeaColors",PlotPoints->36]/.GraphicsComplex[pnts_,data__]:>GraphicsComplex[#[[1]]*{Cos[#[[2]]],Sin[#[[2]]]}&/@pnts,data],PlotRange->All,Frame->False],{j,0,2},{k,0,3}]]

Grid[Table[ParametricPlot3D[{r Cos[t],r Sin[t],BesselJ[j,r*z[[j+1]][[k+1]]]*Cos[j*t]},{r,0,1},{t,-Pi,Pi},ColorFunction->"DeepSeaColors",Axes->False,Boxed->False,Mesh->None],{j,0,2},{k,0,3}]]

Grid[Table[Show[DensityPlot[BesselJ[j,r*z[[j+1]][[k+1]]]*Cos[j*theta],{r,0,1},{theta,-Pi,Pi},ColorFunction->"DeepSeaColors",PlotPoints->36]/.GraphicsComplex[pnts_,data__]:>GraphicsComplex[#[[1]]*{Cos[#[[2]]],Sin[#[[2]]]}&/@pnts,data],PlotRange->All,Frame->False],{k,0,2},{j,0,3}]]

Grid[Table[ParametricPlot3D[{r Cos[t],r Sin[t],BesselJ[j,r*z[[j+1]][[k+1]]]*Cos[j*t]},{r,0,1},{t,-Pi,Pi},ColorFunction->"DeepSeaColors",Axes->False,Boxed->False,Mesh->None],{k,0,2},{j,0,3}]]

Grid[Table[Show[DensityPlot[BesselJ[j,r*z[[j+1]][[k+1]]]*Sin[j*theta],{r,0,1},{theta,-Pi,Pi},ColorFunction->"DeepSeaColors",PlotPoints->36]/.GraphicsComplex[pnts_,data__]:>GraphicsComplex[#[[1]]*{Cos[#[[2]]],Sin[#[[2]]]}&/@pnts,data],PlotRange->All,Frame->False],{j,1,3},{k,0,3}]]

Grid[Table[ParametricPlot3D[{r Cos[t],r Sin[t],BesselJ[j,r*z[[j+1]][[k+1]]]*Sin[j*t]},{r,0,1},{t,-Pi,Pi},ColorFunction->"DeepSeaColors",Axes->False,Boxed->False,Mesh->None],{j,1,3},{k,0,3}]]

Plot[Evaluate@Append[Table[BesselJ[n,x]+n+0.1,{n,1,9}],BesselJ[0,x]],{x,0,50},Filling->Append[Table[n->n+0.1,{n,1,9}],10->Axis],Axes->{True,False}]

50/16//N

DiscretePlot[{BesselJZero[n,1],BesselJZero[n,2]},{n,0,100}]

DiscretePlot[BesselJZero[n,1]/n,{n,100,200}]

DiscretePlot[BesselJZero[n,n+1]-BesselJZero[n,n]-Pi,{n,0,100}]

DiscretePlot[BesselJZero[0,n]/n,{n,100,200}]

k=50;

DiscretePlot[Table[BesselJZero[k,n+1]-BesselJZero[k,n],{k,1,20}],{n,1,20},Filling->None]

DiscretePlot[BesselJZero[k,n+1]-BesselJZero[k,n],{n,1,20},Filling->None]

Plot[{BesselJ[0,z[[1]][[1]]x],BesselJ[0,z[[1]][[2]]x],BesselJ[0,z[[1]][[3]]x],BesselJ[0,z[[1]][[4]]x]},{x,0,1},PlotStyle->Thick,Filling->Axis]

Plot[{BesselJ[1,z[[2]][[1]]x],BesselJ[1,z[[2]][[2]]x],BesselJ[1,z[[2]][[3]]x],BesselJ[1,z[[2]][[4]]x]},{x,0,1},PlotStyle->Thick,Filling->Axis]

Plot[{BesselJ[2,z[[3]][[1]]x],BesselJ[2,z[[3]][[2]]x],BesselJ[2,z[[3]][[3]]x],BesselJ[2,z[[3]][[4]]x]},{x,0,1},PlotStyle->Thick,Filling->Axis]

Plot[{BesselJ[3,z[[4]][[1]]x],BesselJ[3,z[[4]][[2]]x],BesselJ[3,z[[4]][[3]]x],BesselJ[3,z[[4]][[4]]x]},{x,0,1},PlotStyle->Thick,Filling->Axis]

Plot[Evaluate@Append[Table[BesselJ[n,z[[n+1]][[k]]x]+n+0.1,{n,1,3},{k,1,4}],Table[BesselJ[0,z[[1]][[k]]x],{k,1,4}]],{x,0,1},Axes->{True,False}]