residues of tangent and cotangent

We will determine the residuesDlmfPlanetmath of the tangentPlanetmathPlanetmathPlanetmath and the cotangent at their poles, which by the entry are simple (

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

  • We get first

    Res(cot; 0)=limz0zcotz=limz0coszsinzz=11= 1. (1)
  • All the poles of cotangent are  nπ  with  n.  Since π 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-nπ=w) directly

    Res(cot;nπ)=limznπ(z-nπ)cotz=limw0wcot(w+nπ)=limw0wcotw= 1.
  • In the parent entry (, the complement formula for the tangent function is derived.  Using it, we can find the residues of tangent at its poles π2+nπ, which are .  For example,

    Res(tan;π2)=limzπ2(z-π2)cot(π2-z)=limw0wcot(-w)=-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
Canonical name ResiduesOfTangentAndCotangent
Date of creation 2013-03-22 18:57:35
Last modified on 2013-03-22 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