  We give a method of solution to the problem of iterating holomorphic
functions to fractional or complex heights. We construct an auxiliary function
from natural iterates of a holomorphic function; the auxiliary function will be
differintegrable and the complex derivatives of the auxiliary function are the
complex iterates of the original holomorphic function. We use Ramanujan's
master theorem as a foundation and apply elementary theorems from complex
analysis to arrive at our result. We provide non-trivial examples of
holomorphic functions iterated to complex heights using these methods. We
derive a closed form expression for what we call bounded analytic
hyper-operators. These hyper-operators share the same recursive structure as
the hyper-operators defined on the natural numbers but are instead analytic.
They form a sequence of operators beginning with addition, multiplication, and
exponentiation. Surprisingly these hyper-operators are bounded by $e$ as they
grow on the real line. We maintain an elementary yet very general discussion of
the problem, as our solutions are specific instances of more general cases.
