Complex analysis: Difference between revisions

{{distinguish|Complexity analysis }}

'''Complex analysis''', traditionally known as the '''theory of functions of a complex variable''', is the branch of [[mathematical analysis]] that investigates functions of a complex variable of [[complex numbers]]. It is helpful in many branches of mathematics, including [[real analysis]], [[algebraic geometry]], [[number theory]], [[analytic combinatorics]], and [[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 engineering|nuclear]], [[aerospace engineering|aerospace]], [[mechanical engineering|mechanical]] and [[electrical engineering]].<ref>{{Cite web|url=https://gateway.newton.ac.uk/event/ofbw51|title=Industrial Applications of Complex Analysis|date=October 30, 2019|access-date=November 20, 2023|website=Newton Gateway to Mathematics}}</ref>

At first glance, complex analysis is the study of [[holomorphic functions]] that are the  [[differentiable function]]s of a complex variable. By contrast with the real case, a holomorphic function is always [[infinitely differentiable]] and equal to the sum of its [[Taylor series]] in some [[neighborhood (mathematics)|neighborhood]] of each point of its [[domain of a function|domain]].

This makes methods and results of complex analysis significantly different from that of real analysis. The study of real analytic functions often needs the power of complex analysis. This is, in particular, the case in [[analytic combinatorics]].

The theory of [[several complex variables]] generalizes one-variable complex function theory to more than one complex dimension. While many of the techniques of a single complex variable are used and generalized in this setting, several complex variables makes use of additional techniques such as [[Banach algebra]]s and [[sheaf theory]]. It is often concerned with questions of interest in [[algebraic geometry]] and [[symmetric spaces]],

A [[complex function]] is a [[function (mathematics)|function]] from [[complex number]]s to complex numbers. In other words, it is a function that has a subset of the complex numbers as a [[Domain of a function|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]].

In other words, a complex function <math>f:\mathbb{C}\to\mathbb{C}</math> may be decomposed into two real-valued functions (<math>u</math>, <math>v</math>) of two real variables (<math>x</math>, <math>y</math>):

A complex [[Function (mathematics)|function]] is [[continuous function|continuous]] if and only if its associated [[vector-valued function]] of two variables is also continuous. However, this identification does not extend to [[differentiability]]. The definition of the [[derivative]] of a complex function is very similar to that of a real function, but the differentiability of the associated real function of two variables does not imply that the derivative of the complex function exists. In particular, if a complex function has a derivative, it has derivatives of every order and equals the sum of its [[Taylor series]] in a [[neighborhood (mathematics)|neighborhood]] of every point of its domain.

It follows that two differentiable functions that are equal in a [[neighborhood (mathematics)|neighborhood]] of a point are equal on the intersection of their domain if the domains are [[connected space|connected]]. The latter property is the basis of the principle of [[analytic continuation]] which allows extending every real or complex [[analytic function]] in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of [[arc (geometry)|curve arc]]s removed. Many basic and [[special functions|special]] complex functions are defined in this way, including the [[exponential function#Complex plane|complex exponential function]], [[complex logarithm|complex logarithm functions]], and [[trigonometric functions#In the complex plane|trigonometric functions]].