Properties of Complex Numbers

Properties of Complex NumbersSwapnil Sunil JainDecember 26, 2006

Properties of Complex Numbers

Conjugate Properties

${(z_1+z_2)}^{*}=z_1^{*}+z_2^{*}$

${(z_1{z}_2)}^{*}=z_1^{*}{z}_2^{*}$

${({\displaystyle \frac{z_1}{z_2}})}^{*}={\displaystyle \frac{z_1^{*}}{z_2^{*}}}$

${(z^n)}^{*}={(z^{*})}^n$

$f(z^{*})=f^{*}(z)$

$z{z}^{*}={|z|}^2$

Re() and Im() Properties

$z=Re(z)=jIm(z)$

$Re(z)={\displaystyle \frac{z+z^{*}}{2}}$

$Im(z)={\displaystyle \frac{z-z^{*}}{2j}}$

$Re(z^{*})=Re(z)$

$Im(z^{*})=-Im(z)$

$Re(z)\le |z|$

$Im(z)\le |z|$

$Re(z_1+z_2)=Re(z_1)+Re(z_2)$

$Im(z_1+z_2)=Im(z_1)+Im(z_2)$

$Re(z)=Im(jz)$

$Im(z)=Re(-jz)$

Abs() and Arg() Properties

$|z|\equiv {\left[Re{(z)}^2+Im{(z)}^2\right]}^{\frac{1}{2}}$

$|z_1{z}_2|=|z_1||z_2|$

$\left|{\displaystyle \frac{z_1}{z_2}}\right|={\displaystyle \frac{|z_1|}{|z_2|}}$

$|z^{*}|=|z|$

$|z_1+z_2|\le |z_1|+|z_2|$

$\text{arg}(z^{*})=-\text{arg}(z)$

$\text{arg}(z_1{z}_2)=\text{arg}(z_1)+\text{arg}(z_2)$

$\text{arg}\left({\displaystyle \frac{z_1}{z_2}}\right)=\text{arg}(z_1)-\text{arg}(z_2)$

Some Tips

${\displaystyle \frac{1}{j}}=-j$

$-j^2=1$

$a^{log(b)}=b^{log(a)}$

$a=e^{ln(a)}$

Power Properties

For $z=r(\cos (\theta )+j\sin (\theta ))$,

$z^n=r^n(\cos (n\theta )+j\sin (n\theta ))$

$z^{\frac{1}{n}}=r^{\frac{1}{n}}\left[\cos ({\displaystyle \frac{\theta +2k\pi }{n}})+j\sin ({\displaystyle \frac{\theta +2k\pi }{n}})\right]\mathit{\hspace{1em}\hspace{1em}}\text{for }k=0,1,\mathrm{\dots },n-1$

Trigonometric and Logarithmic Properties

$\ln (z)=\ln (|z|e^{j\text{arg}(z)})\equiv \ln (|z|)+j\text{arg}(z)=\ln (|z|)+j(\theta +2k\pi )$

$\exp (z)=\exp (Re(z))\left[\cos (Im(z))+j\sin (Im(z))\right]$

$\cos (z)={\displaystyle \frac{1}{2}}(e^{jz}+e^{-jz})$

$\sin (z)={\displaystyle \frac{1}{2j}}(e^{jz}-e^{-jz})$

$\arccos (z)=-j\ln (z+j{(1-{|z|}^2)}^{\frac{1}{2}})$

$\arcsin (z)=-j\ln (jz+{(1-{|z|}^2)}^{\frac{1}{2}})$

$\cosh (z)={\displaystyle \frac{1}{2}}(e^z+e^{-z})$

$\sinh (z)={\displaystyle \frac{1}{2}}(e^z-e^{-z})$

$\text{arccosh}(z)=-j\ln (z+j{({|z|}^2-1)}^{\frac{1}{2}})$

$\text{arcsinh}(z)=-j\ln (z+{({|z|}^2+1)}^{\frac{1}{2}})$