Integral representations of holomorphic functions

The integral representations for holomorphic functions in a disk D are derived. The integral representations play an important role in complex analysis and the theory of the analytic functions of one complex variable is constructed on the basis of the Cauchy integral formula. The integral representations express the value of the function at every point of D in terms of the values of the function on any circle. The holomorphic functions in D are represented in D by an absolutely converging power series. The absolute convergence of series in D, implies that the series converges uniformly on the interval. Cauchy formula proves that the function is holomorphic in D and it is also holomorphic in the closed disk.