This is a summation of research done in the author’s second and third year of undergraduate mathematics at The University of Toronto. As the previous details were largely scattered and disorganized; the author decided to rewrite the cumulative research. The goal of this paper is to construct a family of analytic functions α ↑ z : (1, e) × CR(z)>0 → CR(z)>0 using methods from fractional calculus. This family satisfies the hyper-operator chain, α ↑ α ↑ z = α ↑ (z + 1); with the initial condition… Expand

This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large… Expand

1. The generalized Mellin transformation The Mellin transformation is a basic tool for analyzing the behavior of many important functions in mathematics and mathematical physics, such as the zeta… Expand

We present a theorem on taking the repeated indefinite summation of a holomorphic function $\phi(z)$ in a vertical strip of $\mathbb{C}$ satisfying exponential bounds as the imaginary part grows. We… Expand

An ideal text for an advanced course in the theory of complex functions, this book leads readers to experience function theory personally and to participate in the work of the creative mathematician.… Expand