complex function

A complex function is a function $f$ from a subset $A$ of $\mathbb{C}$ to $\mathbb{C}$.

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

 $\displaystyle f(z)\;=\;u(x,y)+iv(x,y)$ (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 $\mathbb{R}^{2}$ 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):

 $u(x,y)\;=\;\mbox{Re}\left(f(z)\right)=\Re\left(f(z)\right)$
 $v(x,y)\;=\;\mbox{Im}\left(f(z)\right)=\Im\left(f(z)\right)$

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