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Complex analysis is the study of complex functions, which are functions that take complex numbers as inputs and output complex numbers. It involves the use of techniques from calculus, algebra, and topology to analyze these functions. In the hierarchy of science, mathematics, and functional analysis, complex analysis fits in by providing a way to study complex functions, which are a type of function that can be used to model many phenomena in the natural world. Complex analysis is a key part of functional analysis because it provides tools for studying functions that take complex numbers as inputs, which is an important class of functions in mathematics and science. It also has applications in fields such as physics, engineering, and economics, where models involving complex numbers are commonly used.