  Blandford-Znajek mechanism, by which the rotational energy of a black hole is
extracted through electromagnetic fields, is one of the promising candidates as
an essential process of the central engine of active compact objects such as
Gamma-Ray Bursts. The only known analytical solution of this mechanism is the
perturbative monopole solution for Kerr parameter a up to the second order
terms. In order to apply Blandford-Znajek mechanism to rapidly rotating black
holes, we try to obtain the perturbation solution up to the fourth order. As a
result, we find that the fourth order terms of the vector potential diverge at
infinity, which implies that the perturbation approach breaks down at large
distance from the black hole. Although there are some uncertainties about the
solution due to the unknown boundary condition at infinity for the fourth order
terms, we can derive the evaluation of the total energy flux extracted from the
black hole up to fourth order of a without any ambiguity. Further more, from
the comparison between the numerical solution that is valid for 0<a<1 and the
fourth order solution, we find that the fourth order solution reproduces the
numerical result better than the second order solution. At the same time, since
the fourth order solution does not match well with numerical result at large a,
we conclude that more higher order terms are required to reproduce the
numerical result.
