Category
page 1Theorems in complex analysis
Euler's formula
mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function
fundamental theorem of algebra
Every polynomial has a real or complex root
Euler's identity
e ^ (πi) + 1 = 0
de Moivre's formula
theorem
Cauchy's integral formula
provides integral formulas for all derivatives of a holomorphic function
Abel's theorem
theorem
Cauchy's integral theorem
theorem
residue theorem
the theorem that complex contour integrals are simply the sums of residues of singularities contained within the contour
Liouville's theorem
theorem in complex analysis that nonconstant entire functions are unbounded
Cauchy–Hadamard theorem
theorem
Morera's theorem
theorem that a continuous complex‐valued function on an open set in the complex plane whose contour integral on any closed piecewise‐𝒞¹ curve vanishes must be holomorphic
Bohr–Mollerup theorem
theorem in complex analysis
Picard theorem
theorem
Riemann mapping theorem
theorem that any non-empty simply connected open proper subset of the complex plane is biholomorphic to the open unit disk
Weierstrass factorization theorem
theorem in complex analysis that entire functions can be factorized according to their zeros
Gauss–Lucas theorem
geometric relation between the roots of a polynomial and those of its derivative
Rouché's theorem
theorem
Casorati–Weierstrass theorem
in complex analysis, the theorem that a holomorphic function arbitrarily close to any complex value in every neighborhood of an essential singularity
Jordan's lemma
theorem in complex analysis
argument principle
mathematical relationship
Schwarz lemma
lemma in complex analysis
Mittag-Leffler's theorem
theorem about the existence of meromorphic functions with prescribed poles
Riemann–Roch theorem
theorem that the Euler characteristic of the sheaf cohomology of a holomorphic line bundle on a Riemann surface equals the degree of the bundle plus half of the Euler characteristic of the surface
Hurwitz's theorem
theorem
identity theorem
theorem that an analytic function is completely determined by its values on a countable subset that contains a converging sequence together with its limit
de Branges's theorem
theorem
maximum modulus principle
theorem that a holomorphic function’s absolute value cannot reach a strict local maximum at the interior of its domain
Harnack's principle
theorem
open mapping theorem
Theorem that holomorphic functions on complex domains are open maps
Montel's theorem
theorem
Lagrange inversion theorem
theorem
Jensen's formula
theorem
Hadamard three-circle theorem
theorem
Hartogs' theorem
equivalence between "separate" analyticity and analyticity in multiple variables
Runge's theorem
theorem
Weierstrass preparation theorem
theorem of analytic functions of several complex variables
Osgood's lemma
proposition in complex analysis
Koebe quarter theorem
theorem
Paley–Wiener theorem
any of a family of theorems relating decay properties of a function or distribution at infinity with analyticity of its Fourier transform
Borel–Carathéodory theorem
about the boundedness of a complex analytic function
Bloch's theorem
Theorem in complex analysis named after André Bloch
Analyticity of holomorphic functions
theorem
Hardy's theorem
Mathematical theorem
complex conjugate root theorem
theorem that complex roots of a real polynomial come in conjugate pairs
Mergelyan's theorem
Phragmén–Lindelöf principle
Mathematical technique in complex analysis
Carathéodory's theorem
theorem in complex analysis that a conformal mapping sending the unit disk to a region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve
Schwarz reflection principle
mathematics principle in complex analysis
Hartogs' extension theorem
theorem that singularities of holomorphic functions of several variables cannot be compactly supported
estimation lemma
Concept in integral mathematics
Ostrowski–Hadamard gap theorem
result on the analytic continuation of lacunary power series
Behnke–Stein theorem
Theorem in mathematics about unions of domains of holomorphy
Schottky's theorem
Carlson's theorem
uniqueness theorem in complex analysis
Routh–Hurwitz theorem
mathematical theorem
Looman–Menchoff theorem