  Based on the novel prescription for the power of a complex number, a new
expression for the eigenfunction of the operator of the third component of the
angular momentum is presented. These functions are normalizable, single valued
and are invariant under the rotations at 2\pi for any, not necessary integer m
- the eigenvalue of the operator of the third component of the angular
momentum. For any real m these functions form an orthonormal set, therefore
they may serve as a quantum mechanical eigenfunctions. The eigenfunctions and
eigenvalues of the operator of the angular momentum operator squared, derived
for the two different prescriptions for the square root are reported. The
normalizable eigenfunctions of the operator of the angular momentum operator
squared are presented in terms of hypergeometric functions, admitting integer
as well as non-integer eigenvalues. It is shown that the purely integer
spectrum is not the most general solution but is just the artifact of a
particular choice of the Legendre functions as the pair of linearly independent
solutions of the eigenvalue problem for the operator of the angular momentum
operator squared.
