23.9: Appendix 23B - Complex Numbers
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A complex number \(z\) can be written as a sum of a real number \(x\) and a purely imaginary number \(iy\) where \(i=\sqrt{-1}\)
\[z=x+i y \nonumber \]
The complex number can be represented as a point in the x-y plane as show in Figure 23B.1.
The complex conjugate \(\bar{z}\) of a complex number z is defined to be
\[\bar{z}=x-i y \nonumber \]
The modulus of a complex number is
\[|z|=(z \bar{z})^{1 / 2}=((x+i y)(x-i y))^{1 / 2}=\left(x^{2}+y^{2}\right)^{1 / 2} \nonumber \]
where we used the fact that \(i^{2}=-1\). The modulus \(|z|\) represents the length of the ray from the origin to the complex number z in Figure 23B.1. Let \(\phi\) denote the angle that the ray with the positive x -axis in Figure 23B.1. Then
\[x=|z| \cos \phi \nonumber \]
\[y=|z| \sin \phi \nonumber \]
Hence the angle \(\phi\) is given by
\[\phi=\tan ^{-1}(y / x) \nonumber \]
The inverse of a complex number is then
\[\frac{1}{z}=\frac{\bar{z}}{z \bar{z}}=\frac{x-i y}{\left(x^{2}+y^{2}\right)} \nonumber \]
The modulus of the inverse is the inverse of the modulus;
\[\left|\frac{1}{z}\right|=\frac{1}{\left(x^{2}+y^{2}\right)^{1 / 2}}=\frac{1}{|z|} \nonumber \]
The sum of two complex numbers, \(z_{1}=x_{1}+i y_{1}\) and \(z_{2}=x_{2}+i y_{2}\) is the complex number
\[z_{3}=z_{1}+z_{2}=\left(x_{1}+x_{2}\right)+i\left(y_{1}+y_{2}\right)=x_{3}+i y_{3} \nonumber \]
where \(x_{3}=x_{1}+x_{2}\), \(y_{3}=y_{1}+y_{2}\). We can represent this by the vector sum in Figure 23B.2,
The product of two complex numbers is given by
\[z_{3}=z_{1} z_{2}=\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)=\left(x_{1} x_{2}-y_{1} y_{2}\right)+i\left(x_{1} y_{2}+x_{2} y_{1}\right)=x_{3}+i y_{3} \nonumber \]
where \(x_{3}=x_{1} x_{2}-y_{1} y_{2}\) and \(y_{3}=x_{1} y_{2}+x_{2} y_{1}\)
One of the most important identities in mathematics is the Euler formula,
\[e^{i \phi}=\cos \phi+i \sin \phi \nonumber \]
This identity follows from the power series representations for the exponential, sine, and cosine functions,
\[e^{i \phi}=\sum_{n=0}^{n=\infty} \frac{1}{n !}(i \phi)^{n}=1+i \phi-\frac{\phi^{2}}{2}-i \frac{\phi^{3}}{3 !}+\frac{\phi^{4}}{4 !}+i \frac{\phi^{5}}{5 !} \ldots \nonumber \]
\[\cos \phi=1-\frac{\phi^{2}}{2}+\frac{\phi^{4}}{4 !}-\ldots \nonumber \]
\[\sin \phi=\phi-\frac{\phi^{3}}{3 !}+\frac{\phi^{5}}{5 !}-\ldots \nonumber \]
We define two projection operators. The first one takes the complex number \(e^{i \phi}\) and gives its real part,
\[\operatorname{Re} e^{i \phi}=\cos \phi \nonumber \]
The second operator takes the complex number \(e^{i \phi}\) and gives its imaginary part, which is the real number
\[\operatorname{Im} e^{i \phi}=\sin \phi \nonumber \]
A complex number \(z=x+i y\) can also be represented as the product of a modulus \(|z|\) and a phase factor \(e^{i \phi}\)
\[z=|z| e^{i \phi} \nonumber \]
The inverse of a complex number is then
\[\frac{1}{z}=\frac{1}{|z| e^{i \phi}}=\frac{1}{|z|} e^{-i \phi} \nonumber \]
where we used the fact that
\[\frac{1}{e^{i \phi}}=e^{-i \phi} \nonumber \]
In terms of modulus and phase, the sum of two complex numbers, \(z_{1}=\left|z_{1}\right| e^{i \phi_{1}}\) and \(z_{2}=\left|z_{2}\right| e^{i \phi_{2}}\), is
\[z_{1}+z_{2}=\left|z_{1}\right| e^{i \phi_{1}}+\left|z_{2}\right| e^{i \phi_{2}} \nonumber \]
A special case of this result is when the phase angles are equal, \(\phi_{1}=\phi_{2}\) then the sum \(z_{1}+z_{2}\) has the same phase factor \(e^{i \phi_{1}}\) as \(z_{1}\) and \(z_{2}\)
\[z_{1}+z_{2}=\left|z_{1}\right| e^{i \phi_{1}}+\left|z_{2}\right| e^{i \phi_{1}}=\left(\left|z_{1}\right|+\left|z_{2}\right|\right) e^{i \phi_{1}} \nonumber \]
The product of two complex numbers, \(z_{1}=\left|z_{1}\right| e^{i \phi_{1}}\), and \(z_{2}=\left|z_{2}\right| e^{i \phi_{2}}\) is
\[z_{1} z_{2}=\left|z_{1}\right| e^{i \phi_{1}}\left|z_{2}\right| e^{i \phi_{2}}=\left|z_{1} \| z_{2}\right| e^{i \phi_{1}+\phi_{2}} \nonumber \]
When the phases are equal, the product does not have the same factor as \(z_{1}\) and \(z_{2}\)
\[z_{1} z_{2}=\left|z_{1}\right| e^{i \phi_{1}}\left|z_{2}\right| e^{i \phi_{1}}=\left|z_{1}\right|\left|z_{2}\right| e^{i 2 \phi_{1}} \nonumber \]