CHEM-E4106 - Electrochemistry D, Lecture, 9.1.2023-21.2.2023
This course space end date is set to 21.04.2023 Search Courses: CHEM-E4106
Kirja
11. Appendices
11.3. Relations of complex numbers
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Note that in this book, the imaginary unit i has been replaced by j because i is reserved for the symbol of electric current density.