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- https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/The_Physics_of_Waves_(Goergi)/01%3A_Harmonic_Oscillation/1.04%3A_Complex_NumbersTo divide a complex number \(z\) by a real number \(r\) is easy, just divide both the real and the imaginary parts by \(r\) to get \(z/r = a/r + ib/r.\) To divide by a complex number, \(z'\), we can u...To divide a complex number \(z\) by a real number \(r\) is easy, just divide both the real and the imaginary parts by \(r\) to get \(z/r = a/r + ib/r.\) To divide by a complex number, \(z'\), we can use the fact that \(z'^* z' = |z'|^2\) is real.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/PHY_2030%3A_General_Physics_II/19%3A_Electric_Current_and_Resistance/19.5%3A_Alternating_CurrentsPhasors are used to analyze electrical systems in sinusoidal steady state and with a uniform angular frequency.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/23%3A_Simple_Harmonic_Motion/23.09%3A_Appendix_23B_-_Complex_Numbers\[z_{3}=z_{1} z_{2}=\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)=\left(x_{1} x_{2}-y_{1} y_{2}\right)+i\left(x_{1} y_{2}+x_{2} y_{1}\right)=x_{3}+i y_{3} \nonumber \] \[z_{1}+z_{2}=\left|z_{1}...\[z_{3}=z_{1} z_{2}=\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)=\left(x_{1} x_{2}-y_{1} y_{2}\right)+i\left(x_{1} y_{2}+x_{2} y_{1}\right)=x_{3}+i y_{3} \nonumber \] \[z_{1}+z_{2}=\left|z_{1}\right| e^{i \phi_{1}}+\left|z_{2}\right| e^{i \phi_{1}}=\left(\left|z_{1}\right|+\left|z_{2}\right|\right) e^{i \phi_{1}} \nonumber \] \[z_{1} z_{2}=\left|z_{1}\right| e^{i \phi_{1}}\left|z_{2}\right| e^{i \phi_{2}}=\left|z_{1} \| z_{2}\right| e^{i \phi_{1}+\phi_{2}} \nonumber \]
- https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_and_Applications_(Staelin)/14%3A_Appendices/14.02%3A_Complex_Numbers_and_Sinusoidal_Representation\[\operatorname{Acos}(\omega \mathrm{t}+\phi)=\mathrm{R}_{\mathrm{e}}\left\{\mathrm{Ae}^{\mathrm{j}(\omega \mathrm{t}+\phi)}\right\}=\mathrm{R}_{\mathrm{e}}\left\{\mathrm{Ae}^{\mathrm{j} \phi} \mathrm...\[\operatorname{Acos}(\omega \mathrm{t}+\phi)=\mathrm{R}_{\mathrm{e}}\left\{\mathrm{Ae}^{\mathrm{j}(\omega \mathrm{t}+\phi)}\right\}=\mathrm{R}_{\mathrm{e}}\left\{\mathrm{Ae}^{\mathrm{j} \phi} \mathrm{e}^{\mathrm{j} \omega \mathrm{t}}\right\}=\mathrm{R}_{\mathrm{e}}\left\{\mathrm{Ae}^{\mathrm{j} \omega \mathrm{t}}\right\}=\mathrm{A}_{\mathrm{r}} \cos \omega \mathrm{t}-\mathrm{A}_{\mathrm{i}} \sin \omega \mathrm{t} \label{B.6} \tag{B.6}\]
- https://phys.libretexts.org/Bookshelves/University_Physics/Physics_(Boundless)/19%3A_Electric_Current_and_Resistance/19.5%3A_Alternating_CurrentsPhasors are used to analyze electrical systems in sinusoidal steady state and with a uniform angular frequency.
