Bessel function of the first kind (SymPy, WMA)
BesselJ[n, z]BesselJ[0, 5.2]
D[BesselJ[n, z], z]
BesselJ[0., 0.]
Plot[BesselJ[0, x], {x, 0, 10}]
The special case of half-integer index is expanded using Rayleigh's formulas:
BesselJ[1/2, x]
Some integrals can be expressed in terms of Bessel functions:
Integrate[Cos[3 Sin[w]], {w, 0, Pi}]