complex function ComplexFunction 2004-09-24 15:00:26 2007-05-30 03:31:39 Definition function real part imaginary part function theory complex analysis % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A {\em 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: $$f(z) = u(x,\,y)+iv(x,\,y)$$ The functions $u$ and $v$ are called the {\em real part} and the {\em imaginary part} of the complex function $f$, respectively. 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 \PMlinkescapetext{branch} of mathematics concerning differentiable complex functions is called {\em function theory} or {\em complex analysis}.