Definitions of real order integrals and derivatives using operator   approach
  The set E of functions f fulfilling some conditions is taken to be the definition domain of s-order integral operator J^s (iterative integral), first for any positive integer s and after for any positive s (fractional, transcendental {\pi} and e). The definition of k-order derivative operator D^k for any positive k (fractional, transcendental {\pi} and e) is derived from the definition of J^s. Some properties of J^s and D^k are given and demonstrated. The method is based on the properties of Euler's gamma and beta functions. 
Raoelina Andriambololona
math.GM