# complex function

A complex function is a function $f$ from a subset $A$ of $\u2102$ to $\u2102$.

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:

$f(z)=u(x,y)+iv(x,y)$ | (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):

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

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

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 |