Holomorphic function

A function is holomorphic at a point when it is complex differentiable throughout some open neighborhood of that point. If a function is holomorphic at every point of an open set, it is called holomorphic on that set; the older term analytic is often used for the same class of functions.¹

Writing z = x + iy and f(z) = u(x,y) + iv(x,y), complex differentiability requires the derivative limit to have the same value regardless of the direction of approach. When the first partial derivatives are continuous, this is equivalent to the Cauchy–Riemann equations: u_x = v_y and u_y = −v_x.² The equations are necessary but require suitable regularity to be a sufficient test.

Holomorphicity gives a function far more regular behavior than ordinary real differentiability. Every holomorphic function is infinitely differentiable and locally represented by a convergent power series, so its values near a point are determined by its derivatives there.¹

The Cauchy integral formula expresses the value of a function, and all of its derivatives, through an integral over a surrounding curve. From this follow the identity theorem, the maximum modulus principle, and Liouville’s theorem: a bounded entire function, meaning one holomorphic on the whole complex plane, must be constant.³ Nonconstant holomorphic maps are also conformal wherever their derivative is nonzero, preserving angles between smooth curves while generally changing lengths and areas.

Polynomials, rational functions away from their poles, the exponential function, and the trigonometric functions are standard examples of holomorphic functions. Sums, products, compositions, and quotients remain holomorphic wherever the quotient’s denominator does not vanish; differentiating a power series term by term is also valid within its radius of convergence.²

Real-valued functions of a complex variable are usually not holomorphic unless they are locally constant. The function f(z) = conjugate(z) illustrates the obstruction: its directional difference quotients disagree, and it fails the Cauchy–Riemann equations. Similarly, absolute value and the real-part function are not complex differentiable on any open region. Holomorphicity is therefore a condition on an open set, not merely on isolated points.

Holomorphic functions connect complex analysis with potential theory because their real and imaginary parts are harmonic wherever the function is holomorphic. Conversely, a harmonic function can locally acquire a harmonic conjugate, producing a holomorphic function on suitable simply connected regions.⁴

The behavior near excluded points creates a useful classification. An isolated singularity may be removable, a pole, or essential; Laurent series distinguish these cases, while the residue records the coefficient of the inverse-first-power term and controls many contour integrals.¹ In several complex variables, separate holomorphicity in each coordinate can imply joint holomorphicity, a phenomenon associated with Hartogs’ theorem. Thus local differentiability in complex analysis leads to results with no straightforward real-variable analogue.

Entire function

A holomorphic function defined on the whole complex plane.

Cauchy–Riemann equations

Partial-differential equations characterizing complex differentiability under appropriate regularity conditions.

Conformal map

A map that preserves angles between curves at points where its complex derivative is nonzero.

Residue

The coefficient of (z − a)⁻¹ in the Laurent expansion about an isolated singularity a.