complex tangent and cotangent
Using the Euler’s formulae (http://planetmath.org/ComplexSineAndCosine), one also can define
Thus the properties of the tangent are easily derived from the corresponding properties of the cotangent.
Because of the identic equation the cosine and sine do not vanish simultaneously, and so their quotient is finite in all finite points of the complex plane except in the zeros () of , where becomes infinite. We shall see that these multiples of are simple poles of .
If one moves from to , then both and change their signs (cf. antiperiodic function), and therefore their quotient remains unchanged. Accordingly, is a period of . But if is an arbitrary period of , we have , and especially gives ; then (1) says that , i.e. . Since the prime period of the complex exponential function is , the last equation is valid only for the values (). Thus we have shown that the prime period of is .
We know that
This result, together with
means that is a simple pole of .
As all meromorphic functions, the cotangent may be expressed as a series with the partial fraction (http://planetmath.org/PartialFractionsOfExpressions) terms of the form , where ’s are the poles — see this entry (http://planetmath.org/ExamplesOfInfiniteProducts).
The real (http://planetmath.org/CmplexFunction) and imaginary parts of tangent and cotangent are seen from the formulae
which may be derived from (1) by substituting ().
- 1 R. Nevanlinna & V. Paatero: Funktioteoria. Kustannusosakeyhtiö Otava. Helsinki (1963).
|Title||complex tangent and cotangent|
|Date of creation||2013-03-22 16:49:56|
|Last modified on||2013-03-22 16:49:56|
|Last modified by||pahio (2872)|