complex tangent and cotangent

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

$$\tan z:=\frac{\sin z}{\cos z},\cot z:=\frac{\cos z}{\sin z}.$$

Using the Euler’s formulae (http://planetmath.org/ComplexSineAndCosine), one also can define

$\tan z:=-i{\displaystyle \frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}},\cot z:=i{\displaystyle \frac{e^{iz}+e^{-iz}}{e^{iz}-e^{-iz}}}.$(1)

The subtraction formulae of cosine and sine (http://planetmath.org/ComplexSineAndCosine) yield an additional between the cotangent and tangent:

$$\cot (\frac{\pi }{2}-z)=\frac{\cos (\frac{\pi }{2}-z)}{\sin (\frac{\pi }{2}-z)}=\frac{\cos \frac{\pi }{2}\cos z+\sin \frac{\pi }{2}\sin z}{\sin \frac{\pi }{2}\cos z-\cos \frac{\pi }{2}\sin z}=\frac{\sin z}{\cos z}=\tan z.$$

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

Because of the identic equation  ${\cos }^{2}z+{\sin }^{2}z=1$  the cosine and sine do not vanish simultaneously, and so their quotient $\cot z$ is finite in all finite points $z$ of the complex plane except in the zeros  $z=n\pi$  ($n=0,\pm 1,\pm 2,\mathrm{\dots }$) of $\sin z$, where $\cot z$ becomes infinite.  We shall see that these multiples of $\pi$ are simple poles of $\cot z$.

If one moves from $z$ to $z+\pi$, then both $\cos z$ and $\sin z$ change their signs (cf. antiperiodic function), and therefore their quotient remains unchanged.  Accordingly, $\pi$ is a period of $\cot z$.  But if $\omega$ is an arbitrary period of $\cot z$, we have  $\cot (z+\omega )=\cot z$,  and especially  $z=0$ gives  $\cot \omega =\mathrm{\infty }$;  then (1) says that $e^{i\omega }=e^{-i\omega }$,  i.e.  $e^{2i\omega }=1$.  Since the prime period of the complex exponential function is $2i\pi$, the last equation is valid only for the values  $\omega =n\pi$  ($n=0,\pm 1,\pm 2,\mathrm{\dots }$).  Thus we have shown that the prime period of $\cot z$ is $\pi$.

We know that

$$\frac{\sin z}{z}=\frac{\sin z-\sin 0}{z}\to \cos 0=1\mathit{\hspace{1em}}\mathrm{as}\mathit{\hspace{1em}}z\to 0;$$

therefore

$$z\cot z=\frac{z}{\sin z}\cdot \cos z\to 1\cdot \cos 0=1\mathit{\hspace{1em}}\mathrm{as}\mathit{\hspace{1em}}z\to 0.$$

This result, together with

$$\cot z\to \mathrm{\infty }\mathit{\hspace{1em}}\mathrm{as}\mathit{\hspace{1em}}z\to 0,$$

means that  $z=0$  is a simple pole of $\cot z$.

Because of the periodicity, $\cot z$ has the simple poles in the points$z=0,\pm \pi ,\pm 2\pi ,\mathrm{\dots }$.  Since one has the derivative

$$\frac{d\cot z}{dz}=-\frac{1}{{\sin }^{2}z},$$

$\cot z$ is holomorphic in all finite points except those poles, which accumulate only to the point  $z=\mathrm{\infty }$.  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 (http://planetmath.org/PartialFractionsOfExpressions) terms of the form $\frac{a_{jk}}{{(z-p_j)}^k}$, where $p_j$’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

$$\tan (x+iy)=\frac{\sin x\cos x+i\sinh y\cosh y}{{\cos }^{2}x+{\sinh }^{2}y},$$

$$\cot (x+iy)=\frac{\sin x\cos x-i\sinh y\cosh y}{{\sin }^{2}x+{\sinh }^{2}y},$$

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