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
Canonical name	PropertiesOfComplexNumbers1
Date of creation	2013-03-11 19:29:52
Last modified on	2013-03-11 19:29:52
Owner	swapnizzle (13346)
Last modified by	(0)
Numerical id	1
Author	swapnizzle (0)
Entry type	Definition