<< Calculus`VectorAnalysis`;

SetCoordinates[Cylindrical[r, fi, z]];
CoordinatesToCartesian[{r, fi, z}]

Jvect = {0, 0, Jz[r]};
Laplacian[Jvect];
Thread[% == 
          I*ω*γ*μ*Jvect];
rce = %[[3]];
Simplify[
        DSolve[rce, Jz[r], r], {μ > 0, γ > 0, 
          r > 0, ω > 0}][[1]]

(((-1))^(3/4)) // N

rce
rce /. 
      Equal[a_, b_] :> 
        Equal[Simplify[(a - b)*r], 0]

Plot[BesselJ[0, (1 + I)*ξr] // Abs, {ξr, 0, 5}, 
      PlotRange -> All]