Complex numbers — numbers of the form a + b·i, where i² = −1 — appeared in the sixteenth century as a strange algebraic trick: Cardano's formula for cubic equations sometimes required taking square roots of negative numbers in intermediate steps, even when the final answer was a real number. For two centuries most mathematicians treated them as a useful fiction. Then, in the nineteenth century, Cauchy and Riemann discovered that complex differentiation is far more rigid than its real counterpart: a function differentiable once in the complex sense is automatically differentiable infinitely many times, equal to its Taylor series, and uniquely determined by its values on a small set. Complex analysis — the study of such functions — turned out to be one of the most beautiful and consequential branches of mathematics, with applications spanning physics, engineering, and pure number theory.
A complex-differentiable (or holomorphic) function f: U → ℂ on an open set U ⊂ ℂ satisfies the Cauchy-Riemann equations: writing f(z) = u(x, y) + i·v(x, y) where z = x + i·y, ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x. From these equations follow the miracles of complex analysis. Holomorphic functions are infinitely differentiable: complex differentiability once implies it forever. Holomorphic functions equal their Taylor series on the largest disk where they are defined — they are analytic. Cauchy's integral theorem: the integral of a holomorphic function around a closed loop is zero. Cauchy's integral formula: a holomorphic function's value at any interior point is determined by its values on the boundary. The residue theorem — a closed-loop integral equals 2πi times the sum of residues at enclosed singularities — is a powerful technique for evaluating real definite integrals via contour integration in the complex plane. Singularities are classified: removable, poles (where 1/f is locally holomorphic), and essential (the wildest behavior, characterized by the Casorati-Weierstrass theorem). Laurent series — power series with negative powers — represent functions with singularities. Conformal mappings (holomorphic with non-vanishing derivative) preserve angles, and the Riemann mapping theorem says any simply connected proper subset of ℂ is conformally equivalent to the unit disk. Analytic continuation extends a function defined locally to the largest possible domain — the Riemann zeta function is the canonical example, originally defined for Re(s) > 1 and extended via continuation to all of ℂ except s = 1.
Quantum mechanics is intrinsically complex: wavefunctions take values in ℂ, and the relative phases between them carry physical content (interference patterns). Engineering uses complex analysis throughout: Laplace and Fourier transforms exploit complex methods, electrical circuit analysis runs on complex impedance, control theory analyzes stability via pole placement in the complex plane. Fluid dynamics: two-dimensional potential flow is exactly complex analysis. Number theory uses complex analysis intensively: the Riemann hypothesis is a statement about the zeros of a complex-analytic function, and the Langlands programme deeply involves complex-analytic modular forms. Conformal field theory in physics — central to string theory and statistical mechanics in 2D — is built on the geometry of conformal maps. The unique rigidity of complex differentiability — a property that has no analog in any other branch of analysis — is what makes complex analysis so disproportionately powerful.