Complex numbers. Raising complex numbers to a power

Recall the necessary information about complex numbers.

Complex number is an expression of the form a + bi, where a, b are real numbers, and i- so-called imaginary unit, the symbol whose square is -1, i.e. i 2 = -1. Number a called real part, and the number b - imaginary part complex number z = a + bi. If a b= 0, then instead of a + 0i write simply a. It can be seen that real numbers are a special case of complex numbers.

Arithmetic operations on complex numbers are the same as on real ones: they can be added, subtracted, multiplied and divided by each other. Addition and subtraction proceed according to the rule ( a + bi) ± ( c + di) = (a ± c) + (b ± d)i, and multiplication - according to the rule ( a + bi) · ( c + di) = (ac – bd) + (ad + bc)i(here it is just used that i 2 = -1). Number = a – bi called complex conjugate to z = a + bi. Equality z · = a 2 + b 2 allows you to understand how to divide one complex number by another (non-zero) complex number:

(For example, .)

Complex numbers have a convenient and visual geometric representation: the number z = a + bi can be represented as a vector with coordinates ( a; b) on the Cartesian plane (or, which is almost the same, a point - the end of the vector with these coordinates). In this case, the sum of two complex numbers is depicted as the sum of the corresponding vectors (which can be found by the parallelogram rule). By the Pythagorean theorem, the length of the vector with coordinates ( a; b) is equal to . This value is called module complex number z = a + bi and is denoted by | z|. The angle that this vector makes with the positive direction of the x-axis (counted counterclockwise) is called argument complex number z and denoted by Arg z. The argument is not uniquely defined, but only up to the addition of a multiple of 2 π radians (or 360°, if you count in degrees) - after all, it is clear that turning through such an angle around the origin will not change the vector. But if the vector of length r forms an angle φ with the positive direction of the x-axis, then its coordinates are equal to ( r cos φ ; r sin φ ). Hence it turns out trigonometric notation complex number: z = |z| (cos(Arg z) + i sin(Arg z)). It is often convenient to write complex numbers in this form, because it greatly simplifies calculations. Multiplication of complex numbers in trigonometric form looks very simple: z one · z 2 = |z 1 | · | z 2 | (cos(Arg z 1+arg z 2) + i sin(Arg z 1+arg z 2)) (when multiplying two complex numbers, their moduli are multiplied and the arguments are added). From here follow De Moivre formulas: z n = |z|n(cos( n(Arg z)) + i sin( n(Arg z))). With the help of these formulas, it is easy to learn how to extract roots of any degree from complex numbers. nth root of z is such a complex number w, what w n = z. It's clear that , And where k can take any value from the set (0, 1, ..., n- one). This means that there is always exactly n roots n th degree from a complex number (on the plane they are located at the vertices of a regular n-gon).

§one. Complex numbers

1°. Definition. Algebraic notation.

Definition 1. Complex numbers called ordered pairs of real numbers and , if the concept of equality is defined for them, the operations of addition and multiplication that satisfy the following axioms:

1) Two numbers
and
equal if and only if
,
, i.e.

2) The sum of complex numbers
and

and equal
, i.e.

3) The product of complex numbers
and
the number is called
and equal, i.e.

The set of complex numbers is denoted C.

Formulas (2),(3) for numbers of the form
take the form

whence it follows that the operations of addition and multiplication for numbers of the form
coincide with addition and multiplication for real numbers a complex number of the form
is identified with a real number .

Complex number
called imaginary unit and denoted , i.e.
Then from (3)

From (2),(3)  which means

Expression (4) is called algebraic notation complex number.

In algebraic form, the operations of addition and multiplication take the form:

The complex number is denoted
,- the real part, is the imaginary part, is a purely imaginary number. Designation:
,
.

Definition 2. Complex number
called conjugate with a complex number
.

Properties of complex conjugation.

1)

2)
.

3) If
, then
.

4)
.

5)
is a real number.

The proof is carried out by direct calculation.

Definition 3. Number
called module complex number
and denoted
.

It's obvious that
, and


. The formulas are also obvious:
and
.

2°. Properties of addition and multiplication operations.

1) Commutativity:
,
.

2) Associativity:,
.

3) Distributivity: .

The proof 1) - 3) is carried out by direct calculations based on similar properties for real numbers.

4)
,
.

5) , C ! , satisfying the equation
. Such

6) ,C, 0, ! :
. Such is found by multiplying the equation by



.

Example. Imagine a complex number
in algebraic form. To do this, multiply the numerator and denominator of the fraction by the conjugate of the denominator. We have:

3°. Geometric interpretation of complex numbers. Trigonometric and exponential form of writing a complex number.

Let a rectangular coordinate system be given on the plane. Then
C one can associate a point on the plane with coordinates
.(see Fig. 1). It is obvious that such a correspondence is one-to-one. In this case, real numbers lie on the abscissa axis, and purely imaginary numbers lie on the ordinate axis. Therefore, the abscissa axis is called real axis, and the y-axis − imaginary axis. The plane on which the complex numbers lie is called complex plane.

Note that and
are symmetrical about the origin, and and are symmetrical with respect to Ox.

Each complex number (i.e., each point on the plane) can be associated with a vector starting at the point O and ending at the point
. The correspondence between vectors and complex numbers is one-to-one. Therefore, the vector corresponding to the complex number , denoted by the same letter

D vector line
corresponding to the complex number
, is equal to
, and
,
.

Using the vector interpretation, one can see that the vector
− sum of vectors and , a
− sum of vectors and
.(see Fig. 2). Therefore, the following inequalities are true:

Along with the length vector we introduce the angle between vector and the Ox axis, counted from the positive direction of the Ox axis: if the count is counterclockwise, then the sign of the angle is considered positive, if clockwise, then negative. This corner is called complex number argument and denoted
. Injection is not defined uniquely, but with precision
…. For
the argument is not defined.

Formulas (6) define the so-called trigonometric notation complex number.

From (5) it follows that if
and
then

From (5)
what by and A complex number is uniquely defined. The converse is not true: namely, by the complex number its module is unique, and the argument , due to (7), − with accuracy
. It also follows from (7) that the argument can be found as a solution to the equation

However, not all solutions to this equation are solutions to (7).

Among all the values ​​of the argument of a complex number, one is chosen, which is called the main value of the argument and is denoted
. Usually the main value of the argument is chosen either in the interval
, or in the interval

In trigonometric form, it is convenient to perform multiplication and division operations.

Theorem 1. Module of the product of complex numbers and is equal to the product of the modules, and the argument is equal to the sum of the arguments, i.e.

, a .

Similarly

,

Proof. Let be ,. Then by direct multiplication we get:

Similarly

.■

Consequence(De Moivre's formula). For
Moivre's formula is valid

P example. Let Find the geometric location of the point
. It follows from Theorem 1 that .

Therefore, to construct it, you must first construct a point , which is the inverse about the unit circle, and then find a point symmetrical to it about the x-axis.

Let be
,those.
Complex number
denoted
, i.e. R the Euler formula is valid

As
, then
,
. From Theorem 1
what about the function
it is possible to work as with an ordinary exponential function, i.e. equalities are true

,
,
.

From (8)
exponential notation complex number

, where
,

Example. .

4°. Roots th power of a complex number.

Consider the equation

Let be
, and the solution of Eq. (9) is sought in the form
. Then (9) takes the form
, whence we find that
,
, i.e.

,
,
.

Thus, equation (9) has roots

Let us show that among (10) there are exactly various roots. Really,

are different, because their arguments are different and differ less than
. Further,
, because
. Similarly
.

Thus, equation (9) for
has exactly roots
located at the vertices of a regular -gon inscribed in a circle of radius centered at T.O.

Thus, it has been proven

Theorem 2. root extraction th power of a complex number
always possible. All root values th degree of located at the top of the correct -gon inscribed in a circle with center at zero and radius
. Wherein,

Consequence. Roots -th degree of 1 are expressed by the formula

.

The product of two roots of 1 is a root, 1 is a root -th degree from unity, root
:
.

Let's start with our favorite square.

Example 9

Squaring a complex number

Here you can go in two ways, the first way is to rewrite the degree as a product of factors and multiply the numbers according to the multiplication rule for polynomials.

The second way is to use the well-known school abbreviated multiplication formula:

For a complex number, it is easy to derive your own abbreviated multiplication formula:

A similar formula can be derived for the square of the difference, as well as for the cube of the sum and the cube of the difference. But these formulas are more relevant for complex analysis problems. What if a complex number needs to be raised to, say, the 5th, 10th, or 100th power? It is clear that in algebraic form it is almost impossible to do such a trick, really, think about how you will solve an example like?

And here the trigonometric form of a complex number comes to the rescue and the so-called De Moivre's formula: If a complex number is represented in trigonometric form, then when it is raised to a natural power, the formula is valid:

Just to disgrace.

Example 10

Given a complex number, find.

What should be done? First you need to represent this number in trigonometric form. Astute readers will notice that we have already done this in Example 8:

Then, according to De Moivre's formula:

God forbid, no need to count on a calculator, but in most cases the angle should be simplified. How to simplify? Figuratively speaking, you need to get rid of extra turns. One revolution is a radian or 360 degrees. Find out how many revolutions we have in the argument. For convenience, we make the fraction correct:, after which it becomes clearly visible that you can reduce one revolution:. I hope everyone understands that this is the same angle.

So the final answer would be:

A separate version of the exponentiation problem is the exponentiation of purely imaginary numbers.

Example 12

Raise complex numbers to powers

Here, too, everything is simple, the main thing is to remember the famous equality.

If the imaginary unit is raised to an even power, then the solution technique is as follows:

If the imaginary unit is raised to an odd power, then we “pin off” one “and”, getting an even power:

If there is a minus (or any real coefficient), then it must first be separated:

Extraction of roots from complex numbers. Quadratic equation with complex roots

Consider an example:

Can't extract the root? If we are talking about real numbers, then it is really impossible. In complex numbers, you can extract the root - you can! More precisely, two root:

Are the found roots really the solution of the equation? Let's check:

Which is what needed to be checked.

An abbreviated notation is often used, both roots are written in one line under the “one comb”:.

These roots are also called conjugate complex roots.

How to extract square roots from negative numbers, I think everyone understands: ,,,, etc. In all cases it turns out two conjugate complex roots.