Zeros distribution of a Mittag-Leffler type entire function with applications to the theory of inverse problems

The work is a general survey. We discuss the problem of the zeros distribution for one particular function of the Mittag-Leffler type, which occurs in the theory of inverse problems for the evolution differential equations. Accurate data about the location of zeros in the complex plane is obtained by analytic tools. An explicit representation of zeros is specified with the use of the special Lambert $W$-function. We present the convenient approximate formulas for the zeros calculation. All results are confirmed by the numerical calculations. The relationship between the obtained relations and the investigation of the natural inverse problem for evolution equation in a Banach space is demonstrated. The conditions for the well-posedness of the problem are expressed in terms of zeros distribution of the studied entire function. As a result, for the inverse problem the convenient sufficient conditions for the existence and uniqueness of solution is proposed and, moreover, a constructive algorithm for the finding of a solution is justified.