Holomorphic Functions

A function \( f: U \subseteq \mathbb{C} \to \mathbb{C} \) is said to be differentiable at \( a \in U \) if the following limit exists:

\[ \lim_{z \to a} \frac{f(z) - f(a)}{z - a} \]

Further, \( f \) is said to be holomorphic on \( U \) if it is differentiable for all \( a \) in \( U \).

Holomorphic functions are quite special. Here are some of their properties:

1. All holomorphic functions are analytic, ie equal to their Taylor series
2. If a holomorphic function is bounded and entire, then it is constant (Liouville's theorem)
3. If \( f \) is entire as well as holomorphic, then it can be represented as a (possibly infinite) product of its zeros (Weierstrass Factorization Theorem)