Complex Numbers

Complex Number

Notation

${\displaystyle Z=A+iB=|Z|\angle \theta ={\sqrt {B^{2}+A^{2}}}\angle Tan^{-1}{\frac {B}{A}}=|Z|e^{i\theta }\,}$

Coordination

 ${\displaystyle z=x+yi}$ . In Rectangular plane
${\displaystyle z=z\angle \theta }$ . In Polar plane
${\displaystyle z=z(\cos \theta +i\sin \theta )}$ . In trigonometry
${\displaystyle z=ze^{j\theta }\,}$ . In Complex plane


Complex conjugate Number

${\displaystyle Z^{*}=A-iB=|Z|\angle -\theta ={\sqrt {B^{2}+A^{2}}}\angle -Tan^{-1}{\frac {B}{A}}=|Z|e^{-i\theta }\,}$

 ${\displaystyle z^{*}=x-yi}$ . In Rectangular plane
${\displaystyle z^{*}=z\angle -\theta }$ . In Polar plane
${\displaystyle z^{*}=z^{*}(\cos \theta -i\sin \theta )}$ . In trigonometry angle
${\displaystyle z^{*}=z^{*}e^{j\theta }\,}$ . In Complex plane


Mathematical Operations

Operation on 2 different complex numbers

 Addition ${\displaystyle (A+iB)+(C+iD)=(A+C)+i(B+D)}$ Subtraction ${\displaystyle (A+iB)-(C+iD)=(A-C)+i(B-D)}$ Multilication ${\displaystyle (A+iB)(C+iD)=(AC+BD)+i(AD+BC)}$ Division ${\displaystyle {\frac {(A+iB)}{(C+iD)}}={\frac {(A+iB)}{(C+iD)}}{\frac {(C+iD)}{(C+iD)}}={\frac {(AC+BD)+i(AD+BC)}{(C+iD)^{2}}}}$

Operation on complex numbers and its conjugate

 Addition ${\displaystyle (A+iB)+(A-iB)=2A}$ Subtraction ${\displaystyle (A+iB)-(A-iB)=i2B}$ Multilication ${\displaystyle (A+iB)(A-iB)=A^{2}-B^{2}}$ Division ${\displaystyle {\frac {(A+iB)}{(A-iB)}}={\frac {(A+iB)}{(A-iB)}}{\frac {(A-iB)}{(A-iB)}}={\frac {A^{2}-B^{2}}{(A-iB)^{2}}}}$

In Polar form

Operation on complex number and its conjugate

${\displaystyle Z\times Z^{*}=|Z|\angle \theta \times |Z|\angle -\theta =|Z|^{2}\angle (\theta -\theta )=|Z|^{2}}$
${\displaystyle {\frac {Z}{Z^{*}}}={\frac {|Z|\angle \theta }{|Z|\angle -\theta }}=1\angle 2\theta }$

Operation on 2 different complex numbers

${\displaystyle Z_{1}\times Z_{2}=|Z_{1}|\angle \theta _{1}\times |Z_{1}|\angle \theta _{2}=|Z_{1}|\times |Z_{2}|\angle (\theta _{1}+\theta _{2})}$
${\displaystyle {\frac {Z_{1}}{Z_{2}}}={\frac {|Z_{1}|\angle \theta _{1}}{|Z_{2}|\angle \theta _{2}}}={\frac {Z_{1}}{Z_{2}}}\angle \theta _{1}-\theta _{2}}$

Complex power

A careful analysis of the power series for the exponential, sine, and cosine functions reveals the marvelous

Euler formula

${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta \,}$


of which there is the famous case (for θ = π):

${\displaystyle e^{i\pi }=-1\,}$

More generally,

${\displaystyle x+yi=r(\cos \theta +i\sin \theta )=re^{i\theta }\,}$


de Moivre's formula

${\displaystyle (\cos(x)+i\sin(x))^{n}=\cos(nx)+i\sin(nx)}$


for any real ${\displaystyle x}$ and integer ${\displaystyle n}$. This result is known as .

Transcendental functions

The higher mathematical functions (often called "transcendental functions"), like exponential, log, sine, cosine, etc., can be defined in terms of power series (Taylor series). They can be extended to handle complex arguments in the completely natural way, so these functions are defined over the complex plane. They are in fact "complex analytic functions". Just about any normal function one can think of can be extended to the complex numbers, and is complex analytic. Since the power series coefficients of the common functions are real, they work naturally with conjugates. For example:

${\displaystyle \sin({\overline {z}})={\overline {\sin(z)}}\,}$
${\displaystyle \log({\overline {z}})={\overline {\log(z)}}\,}$

Summary

Complex number

 ${\displaystyle z=x+yi}$ . In Rectangular plane
${\displaystyle z=z\angle \theta }$ . In Polar plane
${\displaystyle z=z(\cos \theta +i\sin \theta )}$ . In trigonometry
${\displaystyle z=ze^{j\theta }\,}$ . In Complex plane


Complex conjugate number

 ${\displaystyle z^{*}=x-yi}$ . In Rectangular plane
${\displaystyle z^{*}=z\angle -\theta }$ . In Polar plane
${\displaystyle z^{*}=z^{*}(\cos \theta -i\sin \theta )}$ . In trigonometry angle
${\displaystyle z^{*}=z^{*}e^{j\theta }\,}$ . In Complex plane