Properties of Complex Numbers
Properties of Complex Numbers Swapnil Sunil Jain December 26, 2006
Properties of Complex Numbers
Conjugate Properties
(z1+z2)*=z*1+z*2 | ||
(z1z2)*=z*1z*2 | ||
(z1z2)*=z*1z*2 | ||
(zn)*=(z*)n | ||
f(z*)=f*(z) | ||
zz*=|z|2 |
Re() and Im() Properties
z=Re(z)=jIm(z) | ||
Re(z)=z+z*2 | ||
Im(z)=z-z*2j | ||
Re(z*)=Re(z) | ||
Im(z*)=-Im(z) | ||
Re(z)≤|z| | ||
Im(z)≤|z| | ||
Re(z1+z2)=Re(z1)+Re(z2) | ||
Im(z1+z2)=Im(z1)+Im(z2) | ||
Re(z)=Im(jz) | ||
Im(z)=Re(-jz) |
Abs() and Arg() Properties
|z|≡[Re(z)2+Im(z)2]12 | ||
|z1z2|=|z1||z2| | ||
|z1z2|=|z1||z2| | ||
|z*|=|z| | ||
|z1+z2|≤|z1|+|z2| | ||
arg(z)≡{arctan(Im(z)Re(z)),x>0arctan(Im(z)Re(z))+π,x<0π2,x=0,y>0-π2,x=0,y<0 | ||
arg(z*)=-arg(z) | ||
arg(z1z2)=arg(z1)+arg(z2) | ||
arg(z1z2)=arg(z1)-arg(z2) |
Some Tips
1j=-j | ||
-j2=1 | ||
alog(b)=blog(a) | ||
a=eln(a) |
Power Properties
For z=r(cos(θ)+jsin(θ)),
zn=rn(cos(nθ)+jsin(nθ)) | ||
z1n=r1n[cos(θ+2kπn)+jsin(θ+2kπn)] |
Trigonometric and Logarithmic Properties
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 |