Holomorphic function

A holomorphic function is a complex-valued function of a complex variable that is complex differentiable at every point in its domain. Formally, a function $f : Ω \to C$ , where $Ω$ is an open subset of the complex plane $C$ , is holomorphic if, for every point $z_{0} \in Ω$ , the limit

f^{'} (z_{0}) = lim_{h \to 0} \frac{f (z_{0} + h) - f (z_{0})}{h}

exists for complex $h$ . This property, known as complex differentiability, implies that the function is not only differentiable but also satisfies the Cauchy-Riemann equations, which are necessary and sufficient conditions for holomorphicity when the function is differentiable.

Key properties of holomorphic functions:

They are infinitely differentiable (smooth) in their domain. See Cauchy integral formula.
They are analytic, meaning they can be represented by a power series in a neighborhood of each point in their domain.
Examples include polynomials, exponential functions $e^{z}$ , and trigonometric functions like $\sin z$ , defined on appropriate domains.
A function satisfies Cauchy-Riemann equations if and only if its Polya vector field has null divergence and null rotational.
The real and imaginary parts of a holomorphic function are harmonic functions.

Key properties: