text: Special Weber's transform  is an important tool in solving problems of the hydrodynamics. It is defined for  as
  where ,  are the Bessel functions of the first and second kind respectively. For  it has a non-trivial kernel which consists of the functions .
  The inverse transform is given by the formula
  Due to non-triviality of the kernel, the inversion identity
  is valid if . Also it is valid in the case of  but only for functions, which are orthogonal to the kernel of  in  with infinitesimal element :