Complex Analysis explores the properties and applications of complex numbers and functions, emphasizing their geometric interpretations and analytical techniques.
A complex function assigns a complex number to every complex input, like \( f(z) = z^2 \) or \( f(z) = e^z \).
A function is analytic (or holomorphic) if it's differentiable at every point in some region. In complex analysis, being differentiable is a much stricter condition than in real analysis and leads to amazing properties.
Analytic functions are incredibly smooth and have derivatives of all orders. They can be represented by power series, and their behavior is tightly controlled by their values in small regions.
Analyticity is the foundation for many magical results in complex analysis, like the Cauchy-Riemann equations and the ability to express complex functions as infinite sums.
\( f(z) = z^2 \) is analytic everywhere in the complex plane.
\( f(z) = |z| \) is not analytic because it doesn't satisfy the necessary conditions.
Analytic functions are smooth and well-behaved, with strong properties stemming from complex differentiability.