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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 branch of mathematics concerning differentiable^{} complex functions is called function theory or complex analysis.

## Mathematics Subject Classification

30A99*no label found*03E20

*no label found*

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