The Theory of Complex Functions

The theory of complex functions studies how functions behave when their inputs and outputs are complex numbers—numbers with real and imaginary parts. It explores functions that are smooth and differentiable in this context, similar to how real functions are studied but with richer structure. These functions exhibit unique properties, like mapping shapes into others while preserving angles and local structures, which leads to deep insights in mathematics and physics. Overall, the theory helps us understand complex systems by analyzing how these special functions transform complex spaces in consistent, predictable ways.