Complex Numbers

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Complex Number[edit]

Notation

Coordination

Complexplane.JPG Euler's formula.svg
  . In Rectangular plane 
  . In Polar plane
  . In trigonometry
  . In Complex plane

Complex conjugate Number[edit]


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

Mathematical Operations[edit]

Operation on 2 different complex numbers[edit]

Addition
Subtraction
Multilication
Division


Operation on complex numbers and its conjugate[edit]

Addition
Subtraction
Multilication
Division

In Polar form[edit]

Operation on complex number and its conjugate

Operation on 2 different complex numbers

Complex power[edit]

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

Euler formula[edit]


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

More generally,



de Moivre's formula[edit]


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

Transcendental functions[edit]

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:

Summary[edit]

Complexplane.JPG Euler's formula.svg

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

References[edit]


See Also[edit]

"Complex Numbers".