**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Category# Complex analysis

Summary

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).
Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Gösta Mittag-Leffler, Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in string theory which examines conformal invariants in quantum field theory.
A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are generally assumed to have a domain that contains a nonempty open subset of the complex plane.
For any complex function, the values from the domain and their images in the range may be separated into real and imaginary parts:
where are all real-valued.
In other words, a complex function may be decomposed into
and
i.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (95)

Related courses (45)

Related categories (127)

Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function.

Tangent half-angle formula

In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following: From these one can derive identities expressing the sine, cosine, and tangent as functions of tangents of half-angles: Using double-angle formulae and the Pythagorean identity gives Taking the quotient of the formulae for sine and cosine yields Combining the Pythagorean identity with the double-angle formula for the cosine, rearranging, and taking the square roots yields and which, upon division gives Alternatively, It turns out that the absolute value signs in these last two formulas may be dropped, regardless of which quadrant α is in.

Elementary function

In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or x1/n). All elementary functions are continuous on their domains. Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-201: Analysis III

Calcul différentiel et intégral.
Eléments d'analyse complexe.

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex

Equations

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true.

Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves.

Topics in analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

Related MOOCs (11)

Related publications (11)

Trigonometric Functions, Logarithms and Exponentials

Ce cours donne les connaissances fondamentales liées aux fonctions trigonométriques, logarithmiques et exponentielles. La présentation des concepts et des propositions est soutenue par une grande gamm

Trigonometric Functions, Logarithms and Exponentials

Ce cours donne les connaissances fondamentales liées aux fonctions trigonométriques, logarithmiques et exponentielles. La présentation des concepts et des propositions est soutenue par une grande gamm

Analyse I

Le contenu de ce cours correspond à celui du cours d'Analyse I, comme il est enseigné pour les étudiantes et les étudiants de l'EPFL pendant leur premier semestre. Chaque chapitre du cours correspond

In this thesis, we give a modern treatment of Dwyer's tame homotopy theory using the language of $\infty$-categories.We introduce the notion of tame spectra and show it has a concrete algebraic description.We then carry out a study of $\infty$-operads and define tame spectral Lie algebras and tame spectral Hopf algebras. Finally, we prove that the homotopy theory of tame spectral Hopf algebras is equivalent to that of tame spaces. To recover Dwyer's Lie algebra model for tame spaces, we use Koszul duality to construct a universal enveloping algebra functor, and show it is an equivalence from the $\infty$-category of tame spectral Lie algebras to the $\infty$-category of tame spectral Hopf algebras.

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.

2020We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space-time white noise or colored noise.

Related lectures (556)

Meromorphic Functions & DifferentialsMATH-410: Riemann surfaces

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

Harmonic Forms and Riemann SurfacesMATH-680: Monstrous moonshine

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Complex Analysis: Holomorphic FunctionsMATH-201: Analysis III

Explores holomorphic functions, Cauchy-Riemann conditions, and principal argument values in complex analysis.