Complex-Order and Fractional Derivatives: A First Exploration

This paper presents an independent exploration of fractional and 
complex-order derivatives, building the framework from algebraic 
first principles using Gamma function extensions. Without prior 
exposure to existing fractional calculus literature, I derive 
formulas for D^α and D^z operators applied to elementary functions, 
prove fundamental properties (linearity, Index Law, product rules), 
and explore geometric interpretations through the "D(i) plane." 
The work concludes with applications to cyclic derivatives and 
preliminary extensions to matrix orders.

Note to Readers: This represents independent rediscovery of 
classical fractional calculus concepts. I have since learned that 
this field has extensive existing literature (Riemann-Liouville, 
Caputo, etc.) and present this work as a pedagogical exercise in 
mathematical exploration rather than novel research.