Theory of General Relativity, as we know, is the geometric theory of gravitation and the current description of gravitation in modern physics. And EinsteinMaxwell equations (also called Einstein field equation), first published by Einstein in 1915 as a tensor equation, describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. Researchers wants to get the solutions of Einstein-Maxwell equations, but unfortunately, because of the special form of the equation, there is no general solution like what we have met in differential equation course. That’s to say, what we can do is to get the solution one by one. many researchers, such as E.T.Newman, W.Kinnersley, M.Demianski, V.S.Manko, E.Ruiz, F.J Ernst, Sibgatullin, etc. have been working on it and has get many kinds of solutions, laying the foundation of further research. It should be pointed out that F.J.Ernst and Sibgatullin’s work provide us a simple and effective method to get the axially symmetric stationary solution, which is what I will show you in this dissertation. Sibgatullin’s integral equations and normalizing conditions are applied to construct general solution of the Einstein-Maxwell field equations whose Ernst complex potentials and corresponding metric functions are obtained explicitly in a simple determinant form. KEY WORDS: axisymmetric , electromagnetic field, gravitational field, complex potentials, Ernst equations, Sibgatullin’s integral, Einstein-Maxwell equations ,