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ALGEBRA
 :: Polar or trigonometric notation of complex numbers

  A point (x, y) of the complex plane that represents the complex number z can also be specified by its distance r from the origin and the angle j between the line joining the point to the origin and the positive x-axis.

Cartesian coordinates expressed by polar coordinates:
x = r cosj
y = r sinj
plugged into  z = x + yi  give
z = r (cosj + isinj),
where

Thus, obtained is the polar or trigonometric form of a complex number where polar coordinates are r, called the absolute value or modulus, and j, that is called the argument, written j = arg(z).

Euler's formula
By using Euler's formula  eij = cosj + isinj,  a complex number can also be written as
z = r eij   which is called the exponential form.
To show the equivalence between the algebraic and the trigonometric form of a complex number,
z = r eij = r (cosj + isinj)
express the sine and the cosine functions in terms of the tangent
and substitute into above expression

Example:  Given the complex number  z = 1 - ÷3i,  express  z = x + yi  in the trigonometric form.

Solution:   The modulus
               the argument
the trigonometric form is
Multiplication and division of complex numbers in the polar form
If given  z1 = r1(cosj1 + isinj1)  and  z2 = r2(cosj2 + isinj2)  then
multiplication:       z1   z2 = r1 r2  [cos(j1 + j2) + isin(j1 + j2)]
division:      
Exponentiation and root extraction of complex numbers in the polar form - de Moivre's formula
We use the polar form for exponentiation and root extraction of complex numbers that are known as de Moivre's formulas.
exponentiation:    zn = rn  [cos(nj) + isin(nj)]
root extraction:   
Powers and roots of complex numbers, use of de Moivreís formulas examples
Example:  Compute
Solution:
or in the polar form,
and
since exponentiation with integer exponent
then
Example:  Compute using de Moivre's formula
Solution:
  since
  then

These complex numbers satisfy the equation z3 = - 8 and by the Fundamental theorem of algebra, since this equation is of degree 3, there must be 3 roots.

Example:  Calculate
Solution:
Eulerís formula, relationship between trigonometric functions and the complex exponential function
Euler's formula
eij = cosj + i sinj,  where
e is the base of the natural logarithm,
i is the imaginary unit, and

j is the angle between x-axis and the vector pointing to the complex number z measured counter clockwise, that is, j is the argument of z,

describes the unit circle in the complex plane.
That is, on the unit circle lie points of the complex plane that correspond to the complex numbers each of which is one unit far from the origin.
Thus, by plugging the angles, into Euler's formula respectively
obtained are the four complex numbers that lie on the unit circle, the two of which lie on the real axis and the two on the imaginary axis as shows the above picture.
The expression  eip + 1 = 0  is called Euler's equation or identity.
Euler's formula shows relations between trigonometric functions and complex exponentials.
Thus, by adding and subtracting Euler's formulas:
                                               (1)   eij = cosj + isinj
               and                           (2)   e-ij = cos(-j) + isin(-j) = cosj - isinj
and solving for cosine and sine
We use Euler's formula to write complex number z = x + yi in polar coordinates,
z = x + yi = r (cosj + isinj) = r eij
 
 
 
 
 
 
 
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