Kirja

Complex number:                  $z=x+iy$, where $i=\sqrt{{-1}}$ is imaginary unit

Complex conjugate:       $z^*=x-iy$

In polar coordinates:         $z=re^{i\varphi}$

                                           $r=\sqrt{zz^*}=\sqrt{x^2+y^2}$

                                           $\varphi=\text{atan}(y/x)$

Euler's formula:                    $re^{\pm i\varphi}=r(\cos\varphi\pm i\sin\varphi)$, where

                                           $x=r\cos(\varphi)$  and  $y=r\sin(\varphi)$

De Moivre's formula:         $(\cos\varphi+i\sin\varphi)^n=\cos(n\varphi)+i\sin(n\varphi)$

Relationship between triginimetric and hyberbolic functions:

$\sin\varphi=\frac{1}{2i}(e^{i\varphi}-e^{-i\varphi})=-i\sinh(i\varphi)$         or       $\sin(i\varphi)=i\sinh(\varphi)$

$\cos\varphi=\frac{1}{2}(e^{i\varphi}-e^{-i\varphi})=\cosh(i\varphi)$            or       $\cos(i\varphi)=\cosh(\varphi)$

$\tan\varphi=\frac{1}{i}\frac{e^{i\varphi}-e^{-i\varphi}}{e^{i\varphi}+e^{-i\varphi}}=-i\tanh(i\varphi)$           or       $\tan(i\varphi)=i\tanh(\varphi)$

$\cot\varphi=\frac{1}{i}\frac{e^{i\varphi}+e^{-i\varphi}}{e^{i\varphi}-e^{-i\varphi}}=-i\coth(i\varphi)$           or        $\cot(i\varphi)=i\coth(\varphi)$       

Exponentials of the imaginary unit:

 $i=e^{i\pi/2} \Rightarrow i^n=(e^{i\pi/2})^n=e^{in\pi/2}=\cos(\frac{n\pi}{n})+i\sin(\frac{n\pi}{2})$

E.g. $\sqrt{i}=i^{1/2}=\cos(\frac{\pi}{4})+i\sin(\frac{\pi}{4})=\frac{1+i}{\sqrt{2}}$

Note that in this book, the imaginary unit i has been replaced by j because i is reserved for the symbol of electric current density.