complex function

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

For every  $z=x+iy\in A(x,y\in ℝ)$  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 ${ℝ}^2$ 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):

$$u(x,y)=\text{Re}\left(f(z)\right)=\mathrm{\Re }\left(f(z)\right)$$

$$v(x,y)=\text{Im}\left(f(z)\right)=\mathrm{\Im }\left(f(z)\right)$$

The of mathematics concerning differentiable 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