Two Forms of the Integral Representations of the Mittag-Leffler Function

The integral representation of the two-parameter Mittag-Leffler function E ρ , μ ( z ) is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex variable to integration over real variables. It is shown that as a result of such a transition, the integral representation of the function E ρ , μ ( z ) has two forms: the representation “A” and “B”. Each of these representations has its advantages and drawbacks. In the paper, the corresponding theorems are formulated and proved, and the advantages and disadvantages of each of the obtained representations are discussed.