CHEM-E4175 - Fundamental Electrochemistry, 09.01.2019-18.02.2019

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Note that in this book, the imaginary unit i has been replaced by j because i is reserved for the symbol of electric current density.