Some useful Identites of complex analysis

let $z$ be a complex number i.e $z=a+i b\;$ , $\;z_1 =a_1+ib_1\;$, $\;z_2 =a_2+ib_2\;$ and $\;|z|=\sqrt{a^{2}+b^{2}}\;$, then we have following properties

$\bullet\;\;|z|=0$ if and only if $z=0$

$\bullet\;\;\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|$

$\bullet\;\;\left|\displaystyle \frac{z_{1}}{z_{2}}\right|=\displaystyle \frac{\left|z_{1}\right|}{\left|z_{2}\right|}$ Provided $\left|z_{2}\right| \neq 0$

$\bullet\;\;\operatorname{Re}\left(z_{1}+z_{2}\right)=\operatorname{Re}\left(z_{1}\right)+\operatorname{Re}\left(z_{2}\right)$

$\bullet\;\;\operatorname{Im}\left(z_{1}+z_{2}\right)=\operatorname{Im}\left(z_{1}\right)+\operatorname{Im}\left(z_{2}\right)$

$\bullet\;\;\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right|$

$\bullet\;\;\left|z_{1}\right|-\left|z_{2}\right||\leq| z_{1}-z_{2} \mid \forall z_{1}, z_{2} \in \mathbb{C}$

$\bullet\;\;\operatorname{Re}(z)=\displaystyle \frac{z+\overline{z}}{2}$

$\bullet\;\;\operatorname{Im}(z)=\displaystyle \frac{z-\overline{z}}{2 i}$

$\bullet\;\;\displaystyle z\overline{z}=|z|^{2}$ or $|\overline{z}|$

$\bullet\;\;$ Given that $z=a+ib$ lets say P is a complex number , then we can represent this in complex plane as a point , the angle $\theta$ line form origin to point P makes with real axis is called argument or amplitude of given point P. can also be expressed as $$\theta=\arctan\left(\frac{b}{a}\right)$$

The argument of a complex number is not unique, lets  say if $\theta$ is argument then $2\pi k +\theta$ is also argument of  point P or $z$.

Principal Argument : When the argument of point $z$ i.e $\theta$ lies between $$-\pi < \theta \leq \pi$$ , then that value of argument is called Principal Argument. denoted  by $\Theta$ or $\operatorname{Arg(z)}$

NOTE : Some authors prefer the range of Principal Argument in $[0, 2 \pi)$

Trigonometric Functions

$\bullet\;\; \cos z =\displaystyle \frac{e^{i z}+e^{-i z}}{2}$

$\bullet\;\; \sin z =\displaystyle \frac{e^{i z}-e^{-i z}}{2}$

$\bullet\;\; \tan z =\displaystyle \frac{\sin z}{\cos z}$

$\bullet\;\; \sin ^{2} z+\cos ^{2} z=1$

$\bullet\;\; 2 \sin z_{1} \cos z_{2}=\sin \left(z_{1}+z_{2}\right)+\sin \left(z_{1}-z_{2}\right)$

$\bullet\;\; 2 \cos z_{1} \cos z_{2}=\cos \left(z_{1}+z_{2}\right)+\cos \left(z_{1}-z_{2}\right)$

$\bullet\;\; 2 \sin z_{1} \sin z_{2}=-\cos \left(z_{1}+z_{2}\right)+\cos \left(z_{1}-z_{2}\right)$

$\bullet\;\; \sin z=0$ iff $z=n \pi \quad(n=0, \pm 1, \ldots)$

$\bullet\;\; \cos z=0$ iff $z=\displaystyle \frac{\pi}{2}+n \pi \quad(n=0, \pm 1, \ldots)$

Hyperbolic functions

$\bullet\;\; \cosh z=\displaystyle \frac{e^{z}+e^{-z}}{2}$

$\bullet\;\; \sinh z=\displaystyle \frac{e^{z}-e^{-z}}{2}$

$\bullet\;\; \cosh ^{2} z-\sinh ^{2}$

$\bullet\;\; \sinh (i z)=i \sin z$

$\bullet\;\; \cosh (i z)=\cos z$

$\bullet\;\; \cosh z=0 \text { iff } z=\left(\displaystyle \frac{\pi}{2}+n \pi\right) i \quad(n=0, \pm 1, \ldots)$

$\bullet\;\; \sinh z=0 \text { iff } z=n \pi i \quad(n=0, \pm 1, \ldots)$

$\bullet\;\; \sinh 0=0, \cosh \theta=1, \tanh 0=0$

$\bullet\;\; \cosh ^{2} z-\sinh ^{2} z=1$

$\bullet\;\; 1-\tanh ^{2} z=\operatorname{sech}^{2} z$

$\bullet\;\; \operatorname{coth}^{2} z-1=\operatorname{cosech}^{2} z$

$\bullet\;\; \sinh (-z)=-\sinh z$

$\bullet\;\; \cosh (-z)=\cosh z$

$\bullet\;\; \tanh (-z)=-\tanh z$

$\bullet\;\; \sinh \left(z_{1} \pm z_{2}\right)=\sinh z_{1} \cosh z_{2} \pm \cosh z_{1} \sinh z_{2}$

$\bullet\;\; \cosh \left(z_{1} \pm z_{2}\right)=\cosh z_{1} \cosh z_{2} \pm \sinh z_{1} \sinh z_{2}$

$\bullet\;\; \tanh \left(z_{3} \pm z_{y}\right)=\displaystyle \frac{\tanh z_{1} \pm \tanh z_{2}}{1 \pm \tanh z_{1} \tanh z_{2}}$

$\bullet\;\; \sin i z=i \sinh z$

$\bullet\;\; \cos i z=\cosh z$

$\bullet\;\; \tan i z=i \tanh z$

$\bullet\;\; \tanh i z=i \tan z$