besselj and bessely are the Bessel functions of the first and second kinds, respectively. They satisfy the Bessel equation:

besseli and besselk are the modified Bessel functions of the first and second kinds, respectively. They satisfy the modified Bessel equation:

besselh are the Bessel functions of the third kind, also known as Hankel functions. They are linear combinations of the preceding Bessel functions:

Note that besselh(v,x) is a synonym for bessel(v,1,x) and besselh[k](v,x) is equivalent to besselh(v,k,x).

By default, these functions will evaluate only when the result is an exact value, or when the input x is a floating point number.  When x is a symbolic expression, they will remain in function form so that they can be manipulated symbolically by themselves or as part of a larger expression.  

$\mathrm{with}\left(\mathrm{MTM}\right)\:$

$\mathrm{diff}\left(\mathrm{besselj}\left(v\,x\right)\,x\right)$

$-\mathrm{BesselJ}\left(v+1\,x\right)+\frac{v\mathrm{BesselJ}\left(v\,x\right)}{x}$

The MTM[besseli], MTM[besselj], MTM[besselk], MTM[bessely] and MTM[besselh] commands were updated in Maple 2021.