# Properties of Complex Numbers

Properties of Complex Numbers Swapnil Sunil Jain December 26, 2006

Properties of Complex Numbers

## Conjugate Properties

 $\displaystyle(z_{1}+z_{2})^{*}=z_{1}^{*}+z_{2}^{*}$ $\displaystyle(z_{1}z_{2})^{*}=z_{1}^{*}z_{2}^{*}$ $\displaystyle(\frac{z_{1}}{z_{2}})^{*}=\frac{z_{1}^{*}}{z_{2}^{*}}$ $\displaystyle(z^{n})^{*}=(z^{*})^{n}$ $\displaystyle f(z^{*})=f^{*}(z)$ $\displaystyle zz^{*}=|z|^{2}$

## Re() and Im() Properties

 $\displaystyle z=Re(z)=jIm(z)$ $\displaystyle Re(z)=\frac{z+z^{*}}{2}$ $\displaystyle Im(z)=\frac{z-z^{*}}{2j}$ $\displaystyle Re(z^{*})=Re(z)$ $\displaystyle Im(z^{*})=-Im(z)$ $\displaystyle Re(z)\leq|z|$ $\displaystyle Im(z)\leq|z|$ $\displaystyle Re(z_{1}+z_{2})=Re(z_{1})+Re(z_{2})$ $\displaystyle Im(z_{1}+z_{2})=Im(z_{1})+Im(z_{2})$ $\displaystyle Re(z)=Im(jz)$ $\displaystyle Im(z)=Re(-jz)$

## Abs() and Arg() Properties

 $\displaystyle|z|\equiv{\Big{[}Re(z)^{2}+Im(z)^{2}\Big{]}}^{\frac{1}{2}}$ $\displaystyle|z_{1}z_{2}|=|z_{1}||z_{2}|$ $\displaystyle\Big{|}\frac{z_{1}}{z_{2}}\Big{|}=\frac{|z_{1}|}{|z_{2}|}$ $\displaystyle|z^{*}|=|z|$ $\displaystyle|z_{1}+z_{2}|\leq|z_{1}|+|z_{2}|$ $\displaystyle\text{arg}(z)\equiv\begin{cases}\arctan\Big{(}\frac{Im(z)}{Re(z)}% \Big{)},&x>0\\ \arctan\Big{(}\frac{Im(z)}{Re(z)}\Big{)}+\pi,&x<0\\ \frac{\pi}{2},&x=0,y>0\\ -\frac{\pi}{2},&x=0,y<0\end{cases}$ $\displaystyle\text{arg}(z^{*})=-\text{arg}(z)$ $\displaystyle\text{arg}(z_{1}z_{2})=\text{arg}(z_{1})+\text{arg}(z_{2})$ $\displaystyle\text{arg}\Big{(}\frac{z_{1}}{z_{2}}\Big{)}=\text{arg}(z_{1})-% \text{arg}(z_{2})$

## Some Tips

 $\displaystyle\frac{1}{j}=-j$ $\displaystyle-j^{2}=1$ $\displaystyle a^{log(b)}=b^{log(a)}$ $\displaystyle a=e^{ln(a)}$

## Power Properties

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

 $\displaystyle z^{n}=r^{n}(\cos(n\theta)+j\sin(n\theta))$ $\displaystyle z^{\frac{1}{n}}=r^{\frac{1}{n}}\Big{[}\cos(\frac{\theta+2k\pi}{n% })+j\sin(\frac{\theta+2k\pi}{n})\Big{]}\qquad\mbox{for }k=0,1,...,n-1$

## Trigonometric and Logarithmic Properties

 $\displaystyle\ln(z)=\ln(|z|e^{j\text{arg}(z)})\equiv\ln(|z|)+j\text{arg}(z)=% \ln(|z|)+j(\theta+2k\pi)$ $\displaystyle\exp(z)=\exp(Re(z))\Big{[}\cos(Im(z))+j\sin(Im(z))\Big{]}$ $\displaystyle\cos(z)=\frac{1}{2}(e^{jz}+e^{-jz})$ $\displaystyle\sin(z)=\frac{1}{2j}(e^{jz}-e^{-jz})$ $\displaystyle\arccos(z)=-j\ln(z+j(1-|z|^{2})^{\frac{1}{2}})$ $\displaystyle\arcsin(z)=-j\ln(jz+(1-|z|^{2})^{\frac{1}{2}})$ $\displaystyle\cosh(z)=\frac{1}{2}(e^{z}+e^{-z})$ $\displaystyle\sinh(z)=\frac{1}{2}(e^{z}-e^{-z})$ $\displaystyle\text{arccosh}(z)=-j\ln(z+j(|z|^{2}-1)^{\frac{1}{2}})$ $\displaystyle\text{arcsinh}(z)=-j\ln(z+(|z|^{2}+1)^{\frac{1}{2}})$
Title Properties of Complex Numbers PropertiesOfComplexNumbers1 2013-03-11 19:29:52 2013-03-11 19:29:52 swapnizzle (13346) (0) 1 swapnizzle (0) Definition