# Category: Theorems in complex analysis

Ushiki's theorem
In mathematics, particularly in the study of functions of several complex variables, Ushiki's theorem, named after S. Ushiki, states that certain well-behaved functions cannot have certain kinds of we
Fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at leas
De Branges's theorem
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the comple
Carathéodory's theorem (conformal mapping)
In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that
Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.
Akhiezer's theorem
In the mathematical field of complex analysis, Akhiezer's theorem is a result about entire functions proved by Naum Akhiezer.
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is complet
Liouville's theorem (complex analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, e
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a
Mittag-Leffler's theorem
In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial
Watson's lemma
In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals.
Carlson's theorem
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions whic
Montel's theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give co
Titchmarsh convolution theorem
The Titchmarsh convolution theorem describes the properties of the support of the convolution of two functions. It was proven by Edward Charles Titchmarsh in 1926.
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential
Grunsky's theorem
In mathematics, Grunsky's theorem, due to the German mathematician Helmut Grunsky, is a result in complex analysis concerning holomorphic univalent functions defined on the unit disk in the complex nu
De Moivre's formula
In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number x and integer n it holds that where i is the imaginary unit (i2 = −1).
Monodromy theorem
In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic functio
Morera's theorem
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic. Morera's theorem states that a con
Arakelyan's theorem
In mathematics, Arakelyan's theorem is a generalization of Mergelyan's theorem from compact subsets of an open subset of the complex plane to relatively closed subsets of an open subset.
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality where e is Euler's number, the base of natural logarithms,i is the imaginary unit, which by definition satisfies i2 =
Corona theorem
In mathematics, the corona theorem is a result about the spectrum of the bounded holomorphic functions on the open unit disc, conjectured by and proved by Lennart Carleson. The commutative Banach alge
Nachbin's theorem
In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provi
Carathéodory kernel theorem
In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin Carathéodory in 1912. The uniform conver
Identity theorem for Riemann surfaces
In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on any subset of its domain that has a limit point
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic map
Siu's semicontinuity theorem
In complex analysis, the Siu semicontinuity theorem implies that the Lelong number of a closed positive current on a complex manifold is semicontinuous. More precisely, the points where the Lelong num
Thom–Sebastiani Theorem
In complex analysis, a branch of mathematics, the Thom–Sebastiani Theorem states: given the germ defined as where are germs of holomorphic functions with isolated singularities, the vanishing cycle co
Picard theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
Sokhotski–Plemelj theorem
The Sokhotski–Plemelj theorem (Polish spelling is Sochocki) is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it is often used in physics, althoug
Rouché's theorem
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region with closed contour , if |g(z)| < |f(z)| on , then f and f + g have
Phragmén–Lindelöf principle
In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an a
In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such
Complex conjugate root theorem
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b real numbers, then its complex conjug
Farrell–Markushevich theorem
In mathematics, the Farrell–Markushevich theorem, proved independently by O. J. Farrell (1899–1981) and A. I. Markushevich (1908–1979) in 1934, is a result concerning the approximation in mean square
Behnke–Stein theorem
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence (i.e., ) of domains of holomorphy is again a domain of holomorphy. It was p
Richards' theorem
Richards' theorem is a mathematical result due to Paul I. Richards in 1947. The theorem states that for, if is a positive-real function (PRF) then is a PRF for all real, positive values of . The theor
Runge's theorem
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following: Den
Edge-of-the-wedge theorem
In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the
Lindelöf's theorem
In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex pl
Weierstrass preparation theorem
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplicat
Wielandt theorem
In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers for which by as the only function defined on the half-plane such that: * is holomorphic on ; *
Residue theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to com
Earle–Hamilton fixed-point theorem
In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into
Lagrange inversion theorem
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.
Bergman–Weil formula
In mathematics, the Bergman–Weil formula is an integral representation for holomorphic functions of several variables generalizing the Cauchy integral formula. It was introduced by and .
Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than
Estimation lemma
In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex-valued, continuous function on the contour Γ and if its absolute va
Vivanti–Pringsheim theorem
The Vivanti–Pringsheim theorem is a mathematical statement in complex analysis, that determines a specific singularity for a function described by certain type of power series. The theorem was origina
Schwarz–Ahlfors–Pick theorem
In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorph
Oka's lemma
In mathematics, Oka's lemma, proved by Kiyoshi Oka, states that in a domain of holomorphy in , the function is plurisubharmonic, where is the distance to the boundary. This property shows that the dom
Classification of Fatou components
In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.
Borel–Carathéodory theorem
In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named f
Bloch's theorem (complex variables)
In complex analysis, a branch of mathematics, Bloch's theorem describes the behaviour of holomorphic functions defined on the unit disk. It gives a lower bound on the size of a disk in which an invers
Looman–Menchoff theorem
In the mathematical field of complex analysis, the Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is holomorphic if and only if it
In mathematical complex analysis, Radó's theorem, proved by Tibor Radó, states that every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is a
Koebe quarter theorem
In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem. The image of an injective analytic function from the unit disk onto a subset of the com
Maximum modulus principle
In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f | cannot exhibit a strict local maximum that is properly within the do
Hurwitz's theorem (complex analysis)
In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with
Hartogs's theorem on separate holomorphicity
In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuo
Andreotti–Norguet formula
The Andreotti–Norguet formula, first introduced by Aldo Andreotti and , is a higher–dimensional analogue of Cauchy integral formula for expressing the derivatives of a holomorphic function. Precisely,
Hardy's theorem
In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let be a holomorphic function on the open ball centered at zero and radius in the comp
Argument principle
In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the functio
Littlewood subordination theorem
In mathematics, the Littlewood subordination theorem, proved by J. E. Littlewood in 1925, is a theorem in operator theory and complex analysis. It states that any holomorphic univalent self-mapping of
Schwarz reflection principle
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic
Bohr–Mollerup theorem
In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for x > 0
Harnack's principle
In the mathematical field of partial differential equations, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality which deals with the convergence of sequences of harmonic f
Fabry gap theorem
In mathematics, the Fabry gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a certain "gap" between them. Such a power series
Grace–Walsh–Szegő theorem
In mathematics, the Grace–Walsh–Szegő coincidence theorem is a result named after John Hilton Grace, Joseph L. Walsh, and Gábor Szegő.
Carleson–Jacobs theorem
In mathematics, the Carleson–Jacobs theorem, introduced by Carleson and, describes the best approximation to a continuous function on the unit circle by a function in a Hardy space.
Weierstrass factorization theorem
In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involvin
Denjoy–Wolff theorem
In mathematics, the Denjoy–Wolff theorem is a theorem in complex analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers i
Schwarz integral formula
In complex analysis, a branch of mathematics, the Schwarz integral formula, named after Hermann Schwarz, allows one to recover a holomorphic function, up to an imaginary constant, from the boundary va
Casorati–Weierstrass theorem
In complex analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is named for Karl Theodor Wilhelm
Identity theorem
In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of or ), i
Wirtinger's representation and projection theorem
In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the repr
Schottky's theorem
In mathematical complex analysis, Schottky's theorem, introduced by Schottky is a quantitative version of Picard's theorem. It states that for a holomorphic function f in the open unit disk that does
Remmert–Stein theorem
In complex analysis, a field in mathematics, the Remmert–Stein theorem, introduced by Reinhold Remmert and Karl Stein, gives conditions for the closure of an analytic set to be analytic. The theorem s
Paley–Wiener theorem
In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymon
Analytic Fredholm theorem
In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and
In complex analysis, a branch of mathematics, theHadamard three-circle theorem is a result about the behavior of holomorphic functions. Let be a holomorphic function on the annulus Let be the maximum
Spijker's lemma
In mathematics, Spijker's lemma is a result in the theory of rational mappings of the Riemann sphere. It states that the image of a circle under a complex rational map with numerator and denominator h
Oka–Weil theorem
In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André
Cauchy's integral theorem
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about lin
Mergelyan's theorem
Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
Jensen's formula
In the mathematical field known as complex analysis, Jensen's formula, introduced by Johan Jensen, relates the average magnitude of an analytic function on a circle with the number of its zeros inside
Branching theorem
In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.
Open mapping theorem (complex analysis)
In complex analysis, the open mapping theorem states that if U is a domain of the complex plane C and f : U → C is a non-constant holomorphic function, then f is an open map (i.e. it sends open subset
Bôcher's theorem
In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher.
Parseval–Gutzmer formula
In mathematics, the Parseval–Gutzmer formula states that, if is an analytic function on a closed disk of radius r with Taylor series then for z = reiθ on the boundary of the disk, which may also be wr
Area theorem (conformal mapping)
In the mathematical theory of conformal mappings, the area theoremgives an inequality satisfied bythe power series coefficients of certain conformal mappings.The theorem is called by that name, not be
Jordan's lemma
In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathem
Oka coherence theorem
In mathematics, the Oka coherence theorem, proved by Kiyoshi Oka, states that the sheaf of germs of holomorphic functions on over a complex manifold is coherent.
Hartogs–Rosenthal theorem
In mathematics, the Hartogs–Rosenthal theorem is a classical result in complex analysis on the uniform approximation of continuous functions on compact subsets of the complex plane by rational functio
Bochner–Martinelli formula
In mathematics, the Bochner–Martinelli formula is a generalization of the Cauchy integral formula to functions of several complex variables, introduced by Enzo Martinelli and Salomon Bochner.
Mellin inversion theorem
In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions underwhich the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defi
Routh–Hurwitz theorem
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynom
König's theorem (complex analysis)
In complex analysis and numerical analysis, König's theorem, named after the Hungarian mathematician Gyula Kőnig, gives a way to estimate simple poles or simple roots of a function. In particular, it
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions wit
Hartogs's extension theorem
In the theory of functions of several complex variables, Hartogs's extension theorem is a statement about the singularities of holomorphic functions of several variables. Informally, it states that th
Osgood's lemma
In mathematics, Osgood's lemma, introduced by William Fogg Osgood, is a proposition in complex analysis. It states that a continuous function of several complex variables that is holomorphic in each v