Complex roots online. Extracting the root of a complex number

numbers in trigonometric form.

Moivre's formula

Let z 1 = r 1 (cos  1 + isin  1) and z 2 = r 2 (cos  2 + isin  2).

The trigonometric notation of a complex number is convenient to use to perform the operations of multiplication, division, raising to an integer power, and extracting the root of power n.

z 1 ∙ z 2 = r 1 ∙ r 2 (cos ( 1 +  2) + i sin ( 1 +  2)).

When multiplying two complex numbers in trigonometric form, their modules are multiplied and their arguments are added. When dividing their modules are divided and the arguments are subtracted.

The consequence of the rule for multiplying a complex number is the rule for raising a complex number to a power.

z = r (cos  + i sin ).

z n = r n (cos n + isin n).

This ratio is called by the Moivre formula.

Example 8.1 Find product and quotient:

and

Solution

z 1 ∙ z 2
∙

=

;

Example 8.2 Write a number in trigonometric form

∙
–I) 7.

Solution

We denote
and z 2 =
- i.

r 1 = | z 1 | = √ 1 2 + 1 2 = √ 2;  1 = arg z 1 = arctan ;

z 1 =
;

r 2 = | z 2 | = √ (√ 3) 2 + (- 1) 2 = 2;  2 = arg z 2 = arctan
;

z 2 = 2
;

z 1 5 = (
) 5
; z 2 7 = 2 7

z = (
) 5 2 7
=

2 9

§ 9 Extracting a root from a complex number

Definition. Rootn-th power of a complex number z (denote
) is a complex number w such that w n = z. If z = 0, then
= 0.

Let z  0, z = r (cos + isin). We denote w =  (cos + sin), then the equation w n = z can be written in the following form

 n (cos (n ) + isin (n )) = r (cos + isin).

Hence  n = r,

 =

Thus, w k =
·
.

There are exactly n different values ​​among these values.

Therefore, k = 0, 1, 2,…, n - 1.

On the complex plane, these points are the vertices of a regular n-gon inscribed in a circle with a radius
centered at point O (Figure 12).

Figure 12

Example 9.1 Find all values
.

Solution.

Let's represent this number in trigonometric form. Let's find its module and argument.

w k =
, where k = 0, 1, 2, 3.

w 0 =
.

w 1 =
.

w 2 =
.

w 3 =
.

On the complex plane, these points are the vertices of a square inscribed in a circle with a radius
centered at the origin (Figure 13).

Picture 13 Picture 14

Example 9.2 Find all values
.

Solution.

z = - 64 = 64 (cos + isin);

w k =
, where k = 0, 1, 2, 3, 4, 5.

w 0 =
; w 1 =
;

w 2 =
w 3 =

w 4 =
; w 5 =
.

On the complex plane, these points are the vertices of a regular hexagon inscribed in a circle of radius 2 centered at point O (0; 0) - Figure 14.

§ 10 Exponential form of a complex number.

Euler's formula

We denote
= cos  + isin  and
= cos  - isin . These ratios are called Euler's formulas .

Function
possesses the usual exponential function properties:

Let the complex number z be written in the trigonometric form z = r (cos + isin).

Using Euler's formula, you can write:

z = r
.

This entry is called exemplary complex number. Using it, we get the rules for multiplication, division, exponentiation and root extraction.

If z 1 = r 1
and z 2 = r 2
?then

z 1 z 2 = r 1 r 2
;

·

z n = r n

, where k = 0, 1,…, n - 1.

Example 10.1 Write a number in algebraic form

z =
.

Solution.

Example 10.2 Solve the equation z 2 + (4 - 3i) z + 4 - 6i = 0.

Solution.

For any complex coefficients, this equation has two roots z 1 and z 1 (possibly coinciding). These roots can be found using the same formula as in the real case. As
takes two values ​​that differ only in sign, then this formula has the form:

Since –9 = 9 · e  i, then the values
there will be numbers:

Then
and
.

Solution.

The sought roots of the equation will be the values
.

For z = –1, we have r = 1, arg (–1) = .

w k =
, k = 0, 1, 2.

Exercises

9 Indicate numbers:

10 Write down the numbers in exponential and algebraic forms:

11 Write down numbers in algebraic and geometric forms:

12 Given numbers

By presenting them in exemplary form, find
.

13 Using the exponential form of a complex number, proceed as follows:

but)
b)

in)
G)

It is impossible to unambiguously extract the root of a complex number, since it has a number of values ​​equal to its power.

Complex numbers are raised to the degree of trigonometric form, for which the Moyward formula is valid:

\ (\ z ^ (k) = r ^ (k) (\ cos k \ varphi + i \ sin k \ varphi), \ forall k \ in N \)

Similarly, this formula is used to calculate the k-root of a complex number (not equal to zero):

\ (\ z ^ (\ frac (1) (k)) = (r (\ cos (\ varphi + 2 \ pi n) + i \ sin (\ varphi + 2 \ pi n))) ^ (\ frac ( 1) (k)) = r ^ (\ frac (1) (k)) \ left (\ cos \ frac (\ varphi + 2 \ pi n) (k) + i \ sin \ frac (\ varphi + 2 \ pi n) (k) \ right), \ forall k> 1, \ forall n \ in N \)

If the complex number is not zero, then the roots of degree k always exist, and they can be represented on the complex plane: they will be the vertices of a k-gon inscribed in a circle centered at the origin and radius \ (\ r ^ (\ frac (1) (k)) \)

Examples of problem solving