Complex Analysis
Complex analysis is the branch of mathematics concerning functions of complex-numbers. It explores properties such as differentiability (holomorphy), integration, and series expansions. A central feature is that complex differentiable functions are necessarily analytic and preserve angles locally, a property known as conformality.
Key Concepts and Applications
- Analytic Continuation: Many functions, like the Riemann zeta function, are initially defined by series on limited domains but can be extended to larger regions of the complex plane while preserving their analytic nature.
- Conformal Mapping: Holomorphic functions with non-zero derivatives act as conformal maps, preserving local angles and shapes. This property allows for the transformation of geometries in ways that are useful in physics (fluid dynamics) and art.
Visualizations in Art
- M.C. Escher’s Print Gallery: Complex Analysis and Conformal Mapping demonstrates how complex analysis underpins M.C. Escher’s 1956 lithograph The Print Gallery.
- The artwork utilizes a logarithmic map to transform the complex plane, creating a spiral structure that allows the viewer to see the scene from multiple scales simultaneously.
- This visualization illustrates the concept of taking the logarithm of an image coordinate system, mapping Euclidean geometry into hyperbolic-like structures via complex transformations 3Blue1Brown.