residues of tangent and cotangent

By the rule in the entry coefficients of Laurent series, in a simple pole   $z=a$  of $f$ one has

 $\mbox{Res}(f;\,a)\;=\;\lim_{z\to a}(z\!-\!a)f(z).$
• We get first

 $\displaystyle\mbox{Res}(\cot;\,0)\;=\;\lim_{z\to 0}z\cot{z}\;=\;\lim_{z\to 0}% \frac{\cos{z}}{\frac{\sin{z}}{z}}\;=\;\frac{1}{1}\;=\;1.$ (1)
• 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

 $\mbox{Res}(\cot;\,n\pi)\;=\;\lim_{z\to n\pi}(z\!-\!n\pi)\cot{z}\;=\;\lim_{w\to 0% }w\cot(w\!+\!n\pi)\;=\;\lim_{w\to 0}w\cot{w}\;=\;1.$
• 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 $\displaystyle\frac{\pi}{2}+n\pi$, which are .  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 $-1$.

Consequently, the residues of cotangent are equal to 1 and the residues of tangent equal to $-1$.

Title residues of tangent and cotangent ResiduesOfTangentAndCotangent 2013-03-22 18:57:35 2013-03-22 18:57:35 pahio (2872) pahio (2872) 6 pahio (2872) Example msc 33B10 msc 30D10 msc 30A99 Residue TechniqueForComputingResidues ResiduesOfGammaFunction