Summary: The present paper is intended for the investigation of the integro-differential equation of the form
with complex and in the space of summable functions on a finite interval of the real axis. Here is the operator of the Riemann-Liouville fractional derivative of complex order and is the function defined by
where, when , coincides with the classical Mittag-Leffler function , and in particular . Thus, when , , , , , , , , and are real numbers, the equation (*) describes the unsaturated behavior of the free electron laser.
The Cauchy-type problem for the above integro-differential equation is considered. It is proved that such a problem is equivalent to the Volterra integral equation of the second kind, and its solution in closed form is established. Special cases are investigated.