residues of tangent and cotangent ResiduesOfTangentAndCotangent 2009-06-10 15:58:13 2009-11-12 20:14:57 Example complex tangent and cotangent % 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 \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} We will determine the residues of the tangent and the cotangent at their poles, which by the \PMlinkid{parent entry}{9074} are \PMlinkname{simple}{SimplePole}. By the rule in the entry coefficients of Laurent series, in a simple pole \,$z = a$\, of $f$ one has $$\mbox{Res}(f;\, a) \;=\; \lim_{z \to a}(z\!-\!a)f(z).$$ \begin{itemize} \item We get first \begin{align} \mbox{Res}(\cot;\,0) \;=\; \lim_{z \to 0}z\cot{z} \;=\; \lim_{z \to 0}\frac{\cos{z}}{\frac{\sin{z}}{z}} \;=\; \frac{1}{1} \;=\;1. \end{align} \item All the poles of cotangent are \,$n\pi$\, with\, $n \in \mathbb{Z}$.\, Since $\pi$ is the period of cotangent, we could infer that the residues in all poles are the same as (1).\, We may also calculate (with the change of variable \,$z\!-\!n\pi = w$) directly $$\mbox{Res}(\cot;\,n\pi) \;=\; \lim_{z \to n\pi}(z\!-\!n\pi)\cot{z} \;=\; \lim_{w \to 0}w\cot(w\!+\!n\pi) \;=\; \lim_{w \to 0}w\cot{w} \;=\; 1.$$ \item In the \PMlinkname{parent entry}{ComplexTangentAndCotangent}, the complement formula for the tangent function is derived.\, Using it, we can find the residues of tangent at its poles $\displaystyle\frac{\pi}{2}+n\pi$, which are \PMlinkescapetext{simple}.\, For example, $$\mbox{Res}(\tan;\,\frac{\pi}{2}) \;=\; \lim_{z \to \frac{\pi}{2}}\left(z\!-\!\frac{\pi}{2}\right)\cot\left(\frac{\pi}{2}\!-\!z\right) \;=\; \lim_{w \to 0}w\cot(-w) \;=\; -\mbox{Res}(\cot;\,0) \;=\; -1.$$ Similarly as above, the residues in other poles are $-1$. \end{itemize} Consequently, the residues of cotangent are equal to 1 and the residues of tangent equal to $-1$.