Complex Numbers glossary

Clear, one-line definitions of the Complex Numbers terms used across the OgbonLab textbooks. Each entry links to the interactive sections where the idea is taught.

analytic

A function is analytic at a point when it equals a convergent power series in some neighborhood; for complex functions, analytic equals holomorphic.

See: Power Series and Analytic Functions, 1D layered medium: PINN vs analytic

argument

For a nonzero complex number z, the angle that z makes with the positive real axis.

cauchy-riemann

The equations uₓ = vᵧ and uᵧ = −vₓ for f = u + iv; satisfied exactly when f is complex-differentiable (holomorphic).

See: The Cauchy-Riemann Equations

conformal

A map between regions of the complex plane is conformal when it preserves angles and orientation; equivalently, holomorphic with nonzero derivative.

See: Conformal Maps and Geometric Function Theory

conjugate

For z = a + bi, the complex conjugate z̄ = a - bi; reflecting z across the real axis.

See: Conjugate priors and analytic posteriors

entire

A complex function holomorphic on all of ℂ; examples include polynomials, exp(z), sin(z), and cos(z).

holomorphic

Complex-differentiable at every point of an open set; equivalently, satisfying the Cauchy-Riemann equations there.

See: Holomorphic Functions and Complex Differentiation

laurent series

A representation of a function as Σ aₙ(z - z₀)ⁿ summed over all integers n; extends power series to allow negative powers.

modulus

For a complex number z = a + bi, the magnitude |z| = √(a² + b²); its distance from the origin.

polar form

A representation of a complex number as r(cos θ + i sin θ), where r is the modulus and θ the argument.

power series

A series Σ aₙ(z - z₀)ⁿ in a variable z; converges inside a disk of radius R, the radius of convergence.

See: Power Series and Analytic Functions

residue

For a function with an isolated singularity at z₀, the coefficient of (z - z₀)⁻¹ in its Laurent expansion; the key quantity in contour integration.

uniformization theorem

Every simply connected Riemann surface is conformally equivalent to one of three models: the Riemann sphere, the complex plane ℂ, or the unit disk.

upper half-plane

The set {z ∈ ℂ : Im(z) > 0}; the standard domain for modular forms and a model of hyperbolic geometry.