complex function

A complex function is a function f from a subset A of to .

For every  z=x+iyA(x,y)  the complex value f(z) can be split into its real and imaginary parts u and v, respectively, which can be considered as real functions of two real variables:

f(z)=u(x,y)+iv(x,y) (1)

The functions u and v are called the real part and the imaginary part of the complex function f, respectively.  Conversely, any two functions u(x,y) and v(x,y) defined in some subset of 2 determine via (1) a complex function f.

If f(z) especially is defined as a polynomial of z, then both u(x,y) and v(x,y) are polynomials of x and y with real coefficients.

Following are the notations for u and v that are used most commonly (the parentheses around f(z) may be omitted):


The of mathematics concerning differentiableMathworldPlanetmathPlanetmath complex functions is called function theory or complex analysis.

Title complex function
Canonical name ComplexFunction
Date of creation 2014-02-23 10:20:21
Last modified on 2014-02-23 10:20:21
Owner Wkbj79 (1863)
Last modified by pahio (2872)
Numerical id 12
Author Wkbj79 (2872)
Entry type Definition
Classification msc 30A99
Classification msc 03E20
Related topic RealFunction
Related topic Meromorphic
Related topic Holomorphic
Related topic Entire
Related topic IndexOfSpecialFunctions
Related topic ValuesOfComplexCosine
Defines real part
Defines imaginary part
Defines function theory
Defines complex analysis