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

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

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

\(\bullet\;\; \cosh ^{2} z-\sinh ^{2} \)

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

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

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

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

\(\bullet\;\; \sinh 0=0, \cosh \theta=1, \tanh 0=0\)

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

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

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

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

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

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

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

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

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

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

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

\(\bullet\;\; \tan i z=i \tanh z\)

\(\bullet\;\; \tanh i z=i \tan z\)