Complex Analysis

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19 domain-specific abstractions whose origin domain is Complex Analysis.

• Blaschke Product — A finite or convergent infinite product of disk-automorphism factors that realizes prescribed zeros while forming an inner analytic function.
• Cauchy's integral formula — A boundary integral that reconstructs every value and derivative of a holomorphic function inside a contour.
• Complex conjugate — Reflect a complex number across the real axis by reversing the sign of its imaginary component, producing an involutive field automorphism that preserves real scalars, sums, products, and modulus.
• Contour integration — Integration of a complex function along an oriented path in the complex plane, with deformation and residues enabling evaluation.
• Domain of holomorphy — A domain in several complex variables that supports a holomorphic function unable to extend across any strictly larger connected domain, equivalently a holomorphically convex natural domain.
• Faber polynomials — Polynomials canonically associated with a normalized Laurent series or conformal map, defined by canceling the principal part of its powers.
• Fundamental theorem of algebra — Every nonconstant one-variable polynomial with complex coefficients has a complex root and therefore factors completely into linear terms.
• Gauss–Lucas theorem — The roots of the derivative of a nonconstant complex polynomial lie in the convex hull of the polynomial's roots.
• Indicator function (complex analysis) — A directional asymptotic function recording an entire function’s exponential growth rate.
• Lacunary value — A complex number omitted from the image of a complex-valued function on its stated domain.
• Meromorphic function — A complex function holomorphic on a domain except at isolated poles, equivalently locally a quotient of holomorphic functions with nonzero denominator.
• Monogenic function — In classical complex analysis, a function possessing a unique finite complex derivative at a point or throughout a stated set.
• Phragmén–Lindelöf principle — A complex-analytic extension of the maximum-modulus principle that controls a holomorphic function on an unbounded domain using boundary bounds plus a growth restriction.
• Plurisubharmonic function — An upper-semicontinuous function on a complex domain whose restriction to every complex line is subharmonic.
• Pseudoanalytic function — A complex-valued function satisfying a generalized first-order Cauchy–Riemann system determined by an admissible coefficient.
• Residue (complex analysis) — The coefficient of the inverse-linear term in a meromorphic function’s Laurent expansion at an isolated singularity.
• Riemann–Hilbert Problem — A complex-analytic boundary-value problem that reconstructs a piecewise holomorphic scalar or matrix function from prescribed multiplicative jumps across an oriented contour, together with normalization and singularity conditions.
• Schwarz triangle function — A conformal map from the upper half-plane onto a curvilinear triangle, expressible as a ratio of hypergeometric solutions with angle parameters.
• Singular integral operators on closed curves — Interpret principal-value Cauchy- and Hilbert-type integrals along a closed curve as boundary operators whose jump relations, projections, and mapping properties encode analytic traces on the curve's two sides.