Editing Complex analysis (section)

{{Short description|Branch of mathematics studying functions of a complex variable}}
{{distinguish|Complexity theory (disambiguation){{!}}Complexity theory}}
{{More footnotes|date=March 2021}}
{{Complex analysis sidebar}}

'''Complex analysis''', traditionally known as the '''theory of functions of a complex variable''', is the branch of [[mathematical analysis]] that investigates [[Function (mathematics)|functions]] of [[complex numbers]]. It is helpful in many branches of mathematics, including [[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>

As a [[differentiable function]] of a complex variable is equal to the [[Function series|sum function]] given by its [[Taylor series]] (that is, it is [[Analyticity of holomorphic functions|analytic]]), complex analysis is particularly concerned with [[analytic function]]s of a complex variable, that is, ''[[holomorphic function]]s''. 
The concept can be extended to [[functions of several complex variables]].

Complex analysis is contrasted with [[real analysis]], which deals with the study of [[real number]]s and [[function of a real variable|functions of a real variable]].