residues of tangent and cotangent
We will determine the residues^{} of the tangent^{} and the cotangent at their poles, which by the http://planetmath.org/node/9074parent entry are simple (http://planetmath.org/SimplePole).
By the rule in the entry coefficients of Laurent series, in a simple pole^{} $z=a$ of $f$ one has
$$\text{Res}(f;a)=\underset{z\to a}{lim}(za)f(z).$$ 

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We get first
$\text{Res}(\mathrm{cot};\mathrm{\hspace{0.17em}0})=\underset{z\to 0}{lim}z\mathrm{cot}z=\underset{z\to 0}{lim}{\displaystyle \frac{\mathrm{cos}z}{\frac{\mathrm{sin}z}{z}}}={\displaystyle \frac{1}{1}}=\mathrm{\hspace{0.33em}1}.$ (1) 
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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 $zn\pi =w$) directly
$$\text{Res}(\mathrm{cot};n\pi )=\underset{z\to n\pi}{lim}(zn\pi )\mathrm{cot}z=\underset{w\to 0}{lim}w\mathrm{cot}(w+n\pi )=\underset{w\to 0}{lim}w\mathrm{cot}w=\mathrm{\hspace{0.33em}1}.$$ 
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In the parent entry (http://planetmath.org/ComplexTangentAndCotangent), the complement formula for the tangent function is derived. Using it, we can find the residues of tangent at its poles $\frac{\pi}{2}}+n\pi $, which are . For example,
$$\text{Res}(\mathrm{tan};\frac{\pi}{2})=\underset{z\to \frac{\pi}{2}}{lim}\left(z\frac{\pi}{2}\right)\mathrm{cot}\left(\frac{\pi}{2}z\right)=\underset{w\to 0}{lim}w\mathrm{cot}(w)=\text{Res}(\mathrm{cot};\mathrm{\hspace{0.17em}0})=1.$$ Similarly as above, the residues in other poles are $1$.
Consequently, the residues of cotangent are equal to 1 and the residues of tangent equal to $1$.
Title  residues of tangent and cotangent 

Canonical name  ResiduesOfTangentAndCotangent 
Date of creation  20130322 18:57:35 
Last modified on  20130322 18:57:35 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  6 
Author  pahio (2872) 
Entry type  Example 
Classification  msc 33B10 
Classification  msc 30D10 
Classification  msc 30A99 
Related topic  Residue 
Related topic  TechniqueForComputingResidues 
Related topic  ResiduesOfGammaFunction 