  The spin-weighted spheroidal equations in the case s=1/2 is thoroughly
studied in the paper by means of the perturbation method in supersymmetry
quantum mechanics. The first-five terms of the super-potential in the series of
the parameter beta are given. The general form of the nth term of the
superpotential is also obtained, which could derived from the previous terms
W_{k}, k<n. From the results, it is easy to give the ground eigenfunction of
the equation. Furthermore, the shape-invariance property is investigated in the
series form of the parameter beta and is proven kept in this series form for
the equations. This nice property guarantee one could obtain the excited
eigenfunctions in the series form from the ground eigenfunctions by the method
in supersymmetry quantum mechanics. This shows the perturbation method method
in supersymmetry quantum mechanics could solve the spin-weight spheroidal wave
equations completely in the series form of the small parameter beta.
