complex tangent and cotangent

The tangentPlanetmathPlanetmathPlanetmath and the cotangent function for complex values of the z are defined with the equations


Using the Euler’s formulae (, one also can define

tanz:=-ieiz-e-izeiz+e-iz,cotz:=ieiz+e-izeiz-e-iz. (1)

The subtraction formulae of cosine and sine ( yield an additional between the cotangent and tangent:


Thus the properties of the tangent are easily derived from the corresponding properties of the cotangent.

Because of the identic equation  cos2z+sin2z=1  the cosine and sine do not vanish simultaneously, and so their quotient cotz is finite in all finite points z of the complex plane except in the zeros  z=nπ  (n=0,±1,±2,) of sinz, where cotz becomes infinite.  We shall see that these multiples of π are simple polesMathworldPlanetmathPlanetmath of cotz.

If one moves from z to z+π, then both cosz and sinz change their signs (cf. antiperiodic functionMathworldPlanetmath), and therefore their quotient remains unchanged.  Accordingly, π is a period of cotz.  But if ω is an arbitrary period of cotz, we have  cot(z+ω)=cotz,  and especially  z=0 gives  cotω=;  then (1) says that  eiω=e-iω,  i.e.  e2iω=1.  Since the prime periodPlanetmathPlanetmathPlanetmath of the complex exponential function is 2iπ, the last equation is valid only for the values  ω=nπ  (n=0,±1,±2,).  Thus we have shown that the prime period of cotz is π.

We know that




This result, together with


means that  z=0  is a simple pole of cotz.

Because of the periodicity, cotz has the simple poles in the points z=0,±π,±2π,.  Since one has the derivative


cotz is holomorphic in all finite points except those poles, which accumulate only to the point  z=.  Thus the cotangent is a meromorphic function.  The same concerns naturally the tangent function.

As all meromorphic functions, the cotangent may be expressed as a series with the partial fraction ( terms of the form ajk(z-pj)k, where pj’s are the poles — see this entry (

The real ( and imaginary partsDlmfMathworld of tangent and cotangent are seen from the formulae


which may be derived from (1) by substituting  z:=x+iy (x,y).


  • 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava. Helsinki (1963).
Title complex tangent and cotangent
Canonical name ComplexTangentAndCotangent
Date of creation 2013-03-22 16:49:56
Last modified on 2013-03-22 16:49:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Definition
Classification msc 30A99
Classification msc 30D10
Classification msc 33B10
Related topic ExamplesOfInfiniteProducts
Related topic QuasiPeriodicFunction
Related topic HyperbolicFunctions
Related topic QuasiperiodicFunction