Holomorphic Function

Overview

Holomorphic functions are the central objects of one-variable complex analysis. Their defining derivative uses complex increments approaching zero from every direction. This requirement is rigid: differentiability on an open region forces local power-series expansions and derivatives of every order.

The resulting theory links differential equations, contour integrals, and approximation. A small amount of local derivative information controls much more of the function than it does in real-variable calculus.

Prerequisites

Know complex arithmetic, limits, and real partial derivatives. Write $z=x+iy$ and a function as $f=u+iv$. Path integrals and power series are needed for the main consequences, though the definition itself uses only a limit.

Definition and conventions

Let $\Omega\subseteq\mathbb C$ be open. A map $f:\Omega\to\mathbb C$ is holomorphic if, for every $a\in\Omega$, the finite complex limit

exists. Being differentiable at one isolated point does not imply holomorphicity near it. In one complex variable, holomorphicity is equivalent to local representation by a convergent power series. This article uses “analytic” in that local power-series sense.

Taylor initially develops the subject with continuously differentiable components; his later Goursat discussion removes that extra assumption. The definition here does not require derivative continuity separately.

Intuition

A complex derivative is one complex multiplier describing the same local change in every direction. Multiplication by a nonzero complex number rotates and scales the plane. General real linear maps can stretch different directions differently; most therefore cannot be complex derivatives.

Examples and non-examples

For $f(z)=z^2$,

so $f$ is holomorphic everywhere. The function $1/z$ is holomorphic on $\mathbb C\setminus\{0\}$.

For complex conjugation at any point, the difference quotient is $\overline h/h$. Real increments give $1$, while purely imaginary increments give $-1$. Thus conjugation is smooth as a real map but nowhere complex differentiable. The function $|z|^2$ is complex differentiable at zero, where its quotient has magnitude $|h|$, but is not holomorphic on any neighborhood of zero.

Important properties

For $C^1$ real components, the Cauchy–Riemann equations $u_x=v_y$ and $u_y=-v_x$ characterize holomorphicity. Without regularity hypotheses, merely checking formal partial-derivative equations is insufficient.

If $f$ is holomorphic near a closed disk and $a$ lies inside it, Cauchy's formula expresses $f(a)$ as a boundary integral. Expanding its kernel geometrically gives the local power series and hence infinite differentiability. The identity theorem then says that two holomorphic functions on a connected domain agreeing on a set with an interior accumulation point agree everywhere. Connectedness and the location of the accumulation point matter.

Connections and further reading

Nonvanishing derivatives give local conformal maps. Isolated singularities lead to Laurent series and residues; holomorphicity on the whole plane defines entire functions. Taylor, §§1–2, 5, and 8 supplies the derivative, integral, and regularity viewpoints.