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

$$\text{Res}(f;a)=\underset{z\to a}{lim}(z-a)f(z).$$

• We get first

  $\text{Res}(\cot ;\mathrm{\hspace{0.17em}0})=\underset{z\to 0}{lim}z\cot z=\underset{z\to 0}{lim}{\displaystyle \frac{\cos z}{\frac{\sin z}{z}}}={\displaystyle \frac{1}{1}}=\mathrm{\hspace{0.33em}1}.$(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

  $$\text{Res}(\cot ;n\pi )=\underset{z\to n\pi }{lim}(z-n\pi )\cot z=\underset{w\to 0}{lim}w\cot (w+n\pi )=\underset{w\to 0}{lim}w\cot w=\mathrm{\hspace{0.33em}1}.$$

• 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,

  $$\text{Res}(\tan ;\frac{\pi }{2})=\underset{z\to \frac{\pi }{2}}{lim}\left(z-\frac{\pi }{2}\right)\cot \left(\frac{\pi }{2}-z\right)=\underset{w\to 0}{lim}w\cot (-w)=-\text{Res}(\cot ;\mathrm{\hspace{0.17em}0})=-1.$$

  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 $-1$.