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 $f:\mathrm{\Omega }\to \mathbb{C}$, where $\mathrm{\Omega }$ is an open subset of the complex plane $\mathbb{C}$, is holomorphic if, for every point $z_0\in \mathrm{\Omega }$, the limit

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. The converse is not true.

Key properties:

• Cauchy--Goursat theorem.
• Cauchy integral formula.
• Liouville theorem (complex analysis): a bounded entire function is constant.
• analytic function