Complex Analysis
Cauchy Integral Formula
A holomorphic function's value at any interior point is determined by its values on the boundary circle.
Residue Theorem
A contour integral of a meromorphic function equals 2πi times the sum of residues of enclosed poles.
Maximum Modulus Principle
A non-constant holomorphic function on a connected open set cannot attain its maximum modulus at an interior point.
Schwarz Lemma
A holomorphic self-map of the unit disk fixing the origin satisfies |f(z)| ≤ |z| and |f'(0)| ≤ 1.
Morera's Theorem
A continuous function whose integral vanishes on every rectangle in a disk is holomorphic.
Identity Theorem (Analytic Continuation)
Two analytic functions on a connected domain that agree on a set with a limit point must be identical.
Open Mapping Theorem (Complex)
A non-constant holomorphic function on a connected open set maps open sets to open sets.
Argument Principle
The winding number of f around 0 along a contour equals the number of zeros minus poles enclosed.
Liouville's Theorem
A bounded entire function on ℂ must be constant — complex differentiability is far more rigid than real differentiability
Fundamental Theorem of Algebra
Every non-constant polynomial with complex coefficients has at least one root in ℂ