Complex Numbers

Complex NumberEdit

Notation

 

Coordination

   
   . In Rectangular plane 
   . In Polar plane
   . In trigonometry
   . In Complex plane

Complex conjugate NumberEdit

 


   . In Rectangular plane 
   . In Polar plane 
   . In trigonometry angle
   . In Complex plane

Mathematical OperationsEdit

Operation on 2 different complex numbersEdit

Addition  
Subtraction  
Multilication  
Division  

Operation on complex numbers and its conjugateEdit

Addition  
Subtraction  
Multiplication  
Division  

In Polar formEdit

Operation on complex number and its conjugate

 
 

Operation on 2 different complex numbers

 
 

Complex powerEdit

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

Euler formulaEdit

 

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

 

More generally,

 


de Moivre's formulaEdit

 

for any real   and integer  . This result is known as .

Transcendental functionsEdit

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:

 
 

SummaryEdit

   

Complex number

   . In Rectangular plane 
   . In Polar plane
   . In trigonometry
   . In Complex plane

Complex conjugate number

   . In Rectangular plane 
   . In Polar plane 
   . In trigonometry angle
   . In Complex plane

ReferencesEdit

See AlsoEdit

"Complex Numbers".