  We present a multidomain spectral approach for Fuchsian ordinary differential
equations in the particular case of the hypergeometric equation. Our hybrid
approach uses Frobenius' method and Moebius transformations in the vicinity of
each of the singular points of the hypergeometric equation, which leads to a
natural decomposition of the real axis into domains. In each domain, solutions
to the hypergeometric equation are constructed via the well-conditioned
ultraspherical spectral method. The solutions are matched at the domain
boundaries to lead to a solution which is analytic on the whole compactified
real line $\mathbb{R}\cup {\infty}$, except for the singular points and cuts of
the Riemann surface on which the solution is defined. The solution is further
extended to the whole Riemann sphere by using the same approach for ellipses
enclosing the singularities. The hypergeometric equation is solved on the
ellipses with the boundary data from the real axis. This solution is continued
as a harmonic function to the interior of the disk by solving the Laplace
equation in polar coordinates with an optimal complexity
Fourier--ultraspherical spectral method.
