What is a Holomorphic Function?
A holomorphic function is a complex function that is differentiable at every point in its domain. These functions are incredibly important in complex analysis, a branch of mathematics that investigates functions of complex numbers.
The beauty of holomorphic functions lies in their property that if they are differentiable at just one point in a domain, they are infinitely differentiable and even analytic within that domain. This can lead to wonderful complex structures and patterns!
Key Features
- Complex differentiability: A key distinction from real differentiability.
- Conformality: Angle-preserving transformations, except at critical points.
- Analytic: These functions possess power series expansions.