  The theory of fractional calculus in the complex plane was not built with a
specific application in mind. The main obstacle to application was the
difficulty with obtaining analytic continuations of fractional derivatives and
integrals. It is known that if a function is analytic in a simply connected
domain, then its fractional derivatives and integrals are also analytic in this
region. In multi-connected domains, in general, this property does not hold.
However, for the set of hypergeometric functions $_{p+1}F_{p},p\geq 2$, this
property is preserved. We propose a new version of fractional calculus, which
allows us to reveal the reason for this preservation, and introduce broad
classes of functions for which an appropriate theory can be constructed. This
allows us to find a meaningful application of our calculus in the asymptotic
theory of linear ODE's with analytic coefficients.
