complex variable analysis
complex variable analysis, branch of mathematics that deals with the calculus of functions of a complex variable, i.e., a variable of the form z=x+iy, where x and y are real and i=1 (see number). A functionw=f(z) of a complex variable z is separable into two parts, w = g_{1}(x,y) + ig_{2}(x,y), where g_{1} and g_{2} are realvalued functions of the real variables x and y. The theory of functions of a complex variable is concerned mainly with functions that have a derivative at every point of a given domain of values for z; such functions are called analytic, regular, or holomorphic. If a function is analytic in a given domain, then it also has continuous derivatives of higher order and can be expanded in an infinite series in terms of these derivatives (i.e., a Taylor's series). The function can also be expressed in the infinite series where z_{0} is a point in the domain. Also of interest in complex variable analysis are the points in a domain, called singular points, where a function fails to have a derivative. The theory of functions of a complex variable was developed during the 19th cent. by A. L. Cauchy, C. F. Gauss, B. Riemann, K. T. Weierstrass, and others.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"complex variable analysis." The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. 19 Sep. 2019 <https://www.encyclopedia.com>.
"complex variable analysis." The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. (September 19, 2019). https://www.encyclopedia.com/reference/encyclopediasalmanacstranscriptsandmaps/complexvariableanalysis
"complex variable analysis." The Columbia Encyclopedia, 6th ed.. . Retrieved September 19, 2019 from Encyclopedia.com: https://www.encyclopedia.com/reference/encyclopediasalmanacstranscriptsandmaps/complexvariableanalysis
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.