\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Needs:
Real Rational Functions ›
Complex Functions ›
Needed by:
Meromorphic Functions ›
Links:
Sheet PDF ›
Graph PDF ›

Complex Rational Functions

Why1

Definition

A complex rational function (or rational function, or fractional function) in $\C $ is a function $f: \C \to \C $ for which there exists polynomials $p: \C \to \C $ and $q: \C \to \C $ in $\C $ so that

\[ f(z) = \frac{p(z)}{q(z)}, \]

for all $z \in \C $. In other words, a rational function is a “quotient” (see  Complex Products) of two polynomials in $\C $.

  1. Future editions will include. ↩︎
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