Cauchy’s integral theorem for the matrix function is studied. Using the iteration formula of the Runge–Kutta fourth-order method, we advance the solution to Cauchy’s integral theorem, wherein Jacobi elliptic integrals and the Mobius transformation are applicable. It is proven that the method converges uniformly locally on the solid triangle. The hyperbolic matrix tangent in the arc length when the contour winds once counterclockwise around the integral is computed. Numerical results have been reported with great success.