Complex analysis

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and integer n it holds that where i is the imaginary unit (i2 = −1).

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 function.

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except for a set of isolated points, which are poles of the function.

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic.

In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit which satisfies the equation i2 = −1.

Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly.

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane.

Rouché's theorem, named after Eugène Rouché, states that if the complex-valued functions f and g are holomorphic inside and on some closed contour K, with |g(z)| < |f(z)| on K, then f and f + g have the same number of zeros inside K, where each zero is counted as many times as its multiplicity.

In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C).

In mathematics, an analytic function is a function that is locally given by a convergent power series.

In mathematics, a conformal map is a function that preserves angles locally.

In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.

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 the circle.

In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.

In complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.

In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation for w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L2, and where μ is a given complex function in L∞(U) of norm less than 1, called the Beltrami coefficient.

In potential theory, a branch of mathematics, Cartan's lemma, named after Henri Cartan, is a bound on the measure and complexity of the set on which a logarithmic Newtonian potential is small.

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series.

In mathematics, the Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems for the Laplacian on bounded domains in Euclidean space.

In mathematics, the Grunsky matrices, or Grunsky operators, are matrices introduced by Grunsky () in complex analysis and geometric function theory.

In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis.