Complex numbers are an extension of the set of real numbers, usually denoted by . Any complex number can be represented as a formal sum , where and are real numbers and is the imaginary unit.
Writing a complex number in the form , , is called the algebraic form of a complex number.
Properties of complex numbers. Geometric interpretation of a complex number.
Actions on complex numbers given in algebraic form:
Let's consider the rules by which arithmetic operations are performed on complex numbers.
If two complex numbers α = a + bi and β = c + di are given, then
α + β = (a + bi) + (c + di) = (a + c) + (b + d)i,
α – β = (a + bi) – (c + di) = (a – c) + (b – d)i. (eleven)
This follows from the definition of the operations of addition and subtraction of two ordered pairs of real numbers (see formulas (1) and (3)). We have received the rules for adding and subtracting complex numbers: in order to add two complex numbers, we must separately add their real parts and, accordingly, their imaginary parts; In order to subtract another from one complex number, it is necessary to subtract their real and imaginary parts, respectively.
The number – α = – a – bi is called the opposite of the number α = a + bi. The sum of these two numbers is zero: - α + α = (- a - bi) + (a + bi) = (-a + a) + (-b + b)i = 0.
To obtain the rule for multiplying complex numbers, we use formula (6), i.e., the fact that i2 = -1. Taking this relation into account, we find (a + bi)(c + di) = ac + adi + bci + bdi2 = ac + (ad + bc)i – bd, i.e.
(a + bi)(c + di) = (ac - bd) + (ad + bc)i . (12)
This formula corresponds to formula (2), which determined the multiplication of ordered pairs of real numbers.
Note that the sum and product of two complex conjugate numbers are real numbers. Indeed, if α = a + bi, = a – bi, then α = (a + bi)(a - bi) = a2 – i2b2 = a2 + b2 , α + = (a + bi) + (a - bi) = ( a + a) + (b - b)i= 2a, i.e.
α + = 2a, α = a2 + b2. (13)
When dividing two complex numbers in algebraic form, one should expect that the quotient is also expressed by a number of the same type, i.e. α/β = u + vi, where u, v R. Let us derive the rule for dividing complex numbers. Let the numbers α = a + bi, β = c + di be given, and β ≠ 0, i.e. c2 + d2 ≠ 0. The last inequality means that c and d do not simultaneously vanish (the case is excluded when c = 0, d = 0). Applying formula (12) and the second of equalities (13), we find:
Therefore, the quotient of two complex numbers is determined by the formula:
corresponding to formula (4).
Using the resulting formula for the number β = c + di, you can find its inverse number β-1 = 1/β. Assuming a = 1, b = 0 in formula (14), we obtain
This formula determines the inverse of a given complex number other than zero; this number is also complex.
For example: (3 + 7i) + (4 + 2i) = 7 + 9i;
(6 + 5i) – (3 + 8i) = 3 – 3i;
(5 – 4i)(8 – 9i) = 4 – 77i;
Operations on complex numbers in algebraic form.
55. Argument of a complex number. Trigonometric form of writing a complex number (derivation).
Arg.com.numbers. – between the positive direction of the real X axis and the vector representing the given number.
Trigon formula. Numbers: ,
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Algebraic form of a complex number.
Addition, subtraction, multiplication and division of complex numbers.
We have already become acquainted with the algebraic form of a complex number - this is the algebraic form of a complex number. Why are we talking about form? The fact is that there are also trigonometric and exponential forms of complex numbers, which will be discussed in the next paragraph.
Operations with complex numbers are not particularly difficult and are not much different from ordinary algebra.
Addition of complex numbers
Example 1
Add two complex numbers,
In order to add two complex numbers, you need to add their real and imaginary parts:
Simple, isn't it? The action is so obvious that it does not require additional comments.
In this simple way you can find the sum of any number of terms: sum the real parts and sum the imaginary parts.
For complex numbers, the first class rule is valid: 
– rearranging the terms does not change the sum.
Subtracting Complex Numbers
Example 2
Find the differences between complex numbers and , if ,
The action is similar to addition, the only peculiarity is that the subtrahend must be put in brackets, and then the parentheses must be opened in the standard way with a change of sign:
The result should not be confusing; the resulting number has two, not three parts. Simply the real part is the compound: . For clarity, the answer can be rewritten as follows: .
Let's calculate the second difference:
Here the real part is also composite:
To avoid any understatement, I will give a short example with a “bad” imaginary part: . Here you can no longer do without parentheses.
Multiplying complex numbers
The time has come to introduce you to the famous equality:
Example 3
Find the product of complex numbers,
Obviously, the work should be written like this:
What does this suggest? It begs to open the brackets according to the rule of multiplication of polynomials. That's what you need to do! All algebraic operations are familiar to you, the main thing is to remember that and be careful.
Let us repeat, omg, the school rule for multiplying polynomials: To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of another polynomial.
I'll write it down in detail:
I hope it was clear to everyone that
Attention, and again attention, most often mistakes are made in signs.
Like the sum, the product of complex numbers is commutable, that is, the equality is true: .
In educational literature and on the Internet, it is easy to find a special formula for calculating the product of complex numbers. Use it if you want, but it seems to me that the approach with multiplying polynomials is more universal and clearer. I won’t give the formula; I think that in this case it’s filling your head with sawdust.
Division of complex numbers
Example 4
Given complex numbers, . Find the quotient.
Let's make a quotient:
The division of numbers is carried out by multiplying the denominator and numerator by the conjugate expression of the denominator.
Let's remember the bearded formula and look at our denominator: . The denominator already has , so the conjugate expression in this case is , that is
According to the rule, the denominator must be multiplied by , and, so that nothing changes, the numerator must be multiplied by the same number:
I'll write it down in detail:
I chose a “good” example: if you take two numbers “from scratch”, then as a result of division you will almost always get fractions, something like .
In some cases, before dividing a fraction, it is advisable to simplify it, for example, consider the quotient of numbers: . Before dividing, we get rid of unnecessary minuses: in the numerator and in the denominator we take the minuses out of brackets and reduce these minuses: 
. For those who like to solve problems, here is the correct answer:
Rarely, but the following task occurs:
Example 5
A complex number is given. Write this number in algebraic form (i.e. in the form).
The technique is the same - we multiply the denominator and numerator by the expression conjugate to the denominator. Let's look at the formula again. The denominator already contains , so the denominator and numerator need to be multiplied by the conjugate expression, that is, by:
In practice, they can easily offer a sophisticated example where you need to perform many operations with complex numbers. No panic: be careful, follow the rules of algebra, the usual algebraic procedure, and remember that .
Trigonometric and exponential form of complex number
In this section we will talk more about the trigonometric form of a complex number. The demonstrative form is much less common in practical tasks. I recommend downloading and, if possible, printing trigonometric tables; methodological material can be found on the page Mathematical formulas and tables. You can't go far without tables.
Any complex number (except zero) can be written in trigonometric form: 
, where is it modulus of a complex number, A - complex number argument. Let's not run away, everything is simpler than it seems.
Let us represent the number on the complex plane. For definiteness and simplicity of explanation, we will place it in the first coordinate quadrant, i.e. we believe that:
Modulus of a complex number is the distance from the origin to the corresponding point in the complex plane. Simply put, module is the length radius vector, which is indicated in red in the drawing.
The modulus of a complex number is usually denoted by: or
Using the Pythagorean theorem, it is easy to derive a formula for finding the modulus of a complex number: . This formula is correct for any meanings "a" and "be".
Note: The modulus of a complex number is a generalization of the concept modulus of a real number, as the distance from a point to the origin.
Argument of a complex number called corner between positive semi-axis the real axis and the radius vector drawn from the origin to the corresponding point. The argument is not defined for singular: .
The principle in question is actually similar to polar coordinates, where the polar radius and polar angle uniquely define the point.
The argument of a complex number is standardly denoted: or
From geometric considerations, we obtain the following formula for finding the argument:
. Attention! This formula only works in the right half-plane! If the complex number is not located in the 1st or 4th coordinate quadrant, then the formula will be slightly different. We will also analyze these cases.
But first, let's look at the simplest examples when complex numbers are located on coordinate axes.
Example 7
Let's make the drawing:
In fact, the task is oral. For clarity, I will rewrite the trigonometric form of a complex number: 
Let us remember firmly, the module – length(which is always non-negative), the argument is corner.
1) Let's represent the number in trigonometric form. Let's find its modulus and argument. It's obvious that . Formal calculation using the formula: .
It is obvious that (the number lies directly on the real positive semi-axis). So the number in trigonometric form is: 
.
The reverse check action is as clear as day:
2) Let us represent the number in trigonometric form. Let's find its modulus and argument. It's obvious that . Formal calculation using the formula: .
Obviously (or 90 degrees). In the drawing, the corner is indicated in red. So the number in trigonometric form is: 
.
Using a table of values of trigonometric functions, it is easy to get back the algebraic form of the number (while also performing a check):
3) Let us represent the number in trigonometric form. Let's find its modulus and argument. It's obvious that . Formal calculation using the formula: .
Obviously (or 180 degrees). In the drawing the corner is indicated in blue. So the number in trigonometric form is: 
.
Examination:
4) And the fourth interesting case. Let's represent the number in trigonometric form. Let's find its modulus and argument. It's obvious that . Formal calculation using the formula: .
The argument can be written in two ways: First way: (270 degrees), and, accordingly: 
. Examination:
However, the following rule is more standard: If the angle is greater than 180 degrees, then it is written with a minus sign and the opposite orientation (“scrolling”) of the angle: (minus 90 degrees), in the drawing the angle is marked in green. It is easy to see that and are the same angle.
Thus, the entry takes the form: 
Attention! In no case should you use the parity of the cosine, the oddness of the sine, and further “simplify” the notation:
By the way, it is useful to remember the appearance and properties of trigonometric and inverse trigonometric functions; reference materials are in the last paragraphs of the page Graphs and properties of basic elementary functions. And complex numbers will be learned much easier!
In the design of the simplest examples, one should write: “it is obvious that the module is equal... it is obvious that the argument is equal to...”. This is really obvious and easy to solve verbally.
Let's move on to consider more common cases. As I already noted, there are no problems with the module; you should always use the formula. But the formulas for finding the argument will be different, it depends on which coordinate quarter the number lies in. In this case, three options are possible (it is useful to copy them down in your notebook):
1) If (1st and 4th coordinate quarters, or right half-plane), then the argument must be found using the formula.
2) If (2nd coordinate quarter), then the argument must be found using the formula 
.
3) If (3rd coordinate quarter), then the argument must be found using the formula 
.
Example 8
Represent complex numbers in trigonometric form: , , , .
Since there are ready-made formulas, it is not necessary to complete the drawing. But there is one point: when you are asked to represent a number in trigonometric form, then It’s better to do the drawing anyway. The fact is that a solution without a drawing is often rejected by teachers; the absence of a drawing is a serious reason for a minus and failure.
Eh, I haven’t drawn anything by hand for a hundred years, here you go:
As always, it turned out a bit dirty =)
I will present the numbers and in complex form, the first and third numbers will be for independent solution.
Let's represent the number in trigonometric form. Let's find its modulus and argument.
Lesson plan.
1. Organizational moment.
2. Presentation of the material.
3. Homework.
4. Summing up the lesson.
During the classes
I. Organizational moment.
II. Presentation of the material.
Motivation.
The expansion of the set of real numbers consists of adding new numbers (imaginary) to the real numbers. The introduction of these numbers is due to the impossibility of extracting the root of a negative number in the set of real numbers.
Introduction to the concept of a complex number.
Imaginary numbers, with which we complement real numbers, are written in the form bi, Where i is an imaginary unit, and i 2 = - 1.
Based on this, we obtain the following definition of a complex number.
Definition. A complex number is an expression of the form a+bi, Where a And b- real numbers. In this case, the following conditions are met:
a) Two complex numbers a 1 + b 1 i And a 2 + b 2 i equal if and only if a 1 =a 2, b 1 =b 2.
b) The addition of complex numbers is determined by the rule:
(a 1 + b 1 i) + (a 2 + b 2 i) = (a 1 + a 2) + (b 1 + b 2) i.
c) Multiplication of complex numbers is determined by the rule:
(a 1 + b 1 i) (a 2 + b 2 i) = (a 1 a 2 - b 1 b 2) + (a 1 b 2 - a 2 b 1) i.
Algebraic form of a complex number.
Writing a complex number in the form a+bi is called the algebraic form of a complex number, where A– real part, bi is the imaginary part, and b– real number.
Complex number a+bi is considered equal to zero if its real and imaginary parts are equal to zero: a = b = 0
Complex number a+bi at b = 0 considered to be the same as a real number a: a + 0i = a.
Complex number a+bi at a = 0 is called purely imaginary and is denoted bi: 0 + bi = bi.
Two complex numbers z = a + bi And = a – bi, differing only in the sign of the imaginary part, are called conjugate.
Operations on complex numbers in algebraic form.
You can perform the following operations on complex numbers in algebraic form.
1) Addition.
Definition. Sum of complex numbers z 1 = a 1 + b 1 i And z 2 = a 2 + b 2 i is called a complex number z, the real part of which is equal to the sum of the real parts z 1 And z 2, and the imaginary part is the sum of the imaginary parts of numbers z 1 And z 2, that is z = (a 1 + a 2) + (b 1 + b 2)i.
Numbers z 1 And z 2 are called terms.
Addition of complex numbers has the following properties:
1º. Commutativity: z 1 + z 2 = z 2 + z 1.
2º. Associativity: (z 1 + z 2) + z 3 = z 1 + (z 2 + z 3).
3º. Complex number –a –bi called the opposite of a complex number z = a + bi. Complex number, opposite of complex number z, denoted -z. Sum of complex numbers z And -z equal to zero: z + (-z) = 0
Example 1: Perform addition (3 – i) + (-1 + 2i).
(3 – i) + (-1 + 2i) = (3 + (-1)) + (-1 + 2) i = 2 + 1i.
2) Subtraction.
Definition. Subtract from a complex number z 1 complex number z 2 z, What z + z 2 = z 1.
Theorem. The difference between complex numbers exists and is unique.
Example 2: Perform a subtraction (4 – 2i) - (-3 + 2i).
(4 – 2i) - (-3 + 2i) = (4 - (-3)) + (-2 - 2) i = 7 – 4i.
3) Multiplication.
Definition. Product of complex numbers z 1 =a 1 +b 1 i And z 2 =a 2 +b 2 i is called a complex number z, defined by the equality: z = (a 1 a 2 – b 1 b 2) + (a 1 b 2 + a 2 b 1)i.
Numbers z 1 And z 2 are called factors.
Multiplication of complex numbers has the following properties:
1º. Commutativity: z 1 z 2 = z 2 z 1.
2º. Associativity: (z 1 z 2)z 3 = z 1 (z 2 z 3)
3º. Distributivity of multiplication relative to addition:
(z 1 + z 2) z 3 = z 1 z 3 + z 2 z 3.
4º. z = (a + bi)(a – bi) = a 2 + b 2- real number.
In practice, multiplication of complex numbers is carried out according to the rule of multiplying a sum by a sum and separating the real and imaginary parts.
In the following example, we will consider multiplying complex numbers in two ways: by rule and by multiplying sum by sum.
Example 3: Do the multiplication (2 + 3i) (5 – 7i).
1 way. (2 + 3i) (5 – 7i) = (2× 5 – 3× (- 7)) + (2× (- 7) + 3× 5)i = = (10 + 21) + (- 14 + 15 )i = 31 + i.
Method 2. (2 + 3i) (5 – 7i) = 2× 5 + 2× (- 7i) + 3i× 5 + 3i× (- 7i) = = 10 – 14i + 15i + 21 = 31 + i.
4) Division.
Definition. Divide a complex number z 1 to a complex number z 2, means to find such a complex number z, What z · z 2 = z 1.
Theorem. The quotient of complex numbers exists and is unique if z 2 ≠ 0 + 0i.
In practice, the quotient of complex numbers is found by multiplying the numerator and denominator by the conjugate of the denominator.
Let z 1 = a 1 + b 1 i, z 2 = a 2 + b 2 i, Then
.
In the following example, we will perform division using the formula and the rule of multiplication by the number conjugate to the denominator.
Example 4. Find the quotient 
.
5) Raising to a positive whole power.
a) Powers of the imaginary unit.
Taking advantage of equality i 2 = -1, it is easy to define any positive integer power of the imaginary unit. We have:
i 3 = i 2 i = -i,
i 4 = i 2 i 2 = 1,
i 5 = i 4 i = i,
i 6 = i 4 i 2 = -1,
i 7 = i 5 i 2 = -i,
i 8 = i 6 i 2 = 1 etc.
This shows that the degree values i n, Where n– a positive integer, periodically repeated as the indicator increases by 4 .
Therefore, to raise the number i to a positive whole power, we must divide the exponent by 4 and build i to a power whose exponent is equal to the remainder of the division.
Example 5: Calculate: (i 36 + i 17) i 23.
i 36 = (i 4) 9 = 1 9 = 1,
i 17 = i 4 × 4+1 = (i 4) 4 × i = 1 · i = i.
i 23 = i 4 × 5+3 = (i 4) 5 × i 3 = 1 · i 3 = - i.
(i 36 + i 17) · i 23 = (1 + i) (- i) = - i + 1= 1 – i.
b) Raising a complex number to a positive integer power is carried out according to the rule for raising a binomial to the corresponding power, since it is a special case of multiplying identical complex factors.
Example 6: Calculate: (4 + 2i) 3
(4 + 2i) 3 = 4 3 + 3× 4 2 × 2i + 3× 4× (2i) 2 + (2i) 3 = 64 + 96i – 48 – 8i = 16 + 88i.
Algebraic form of writing a complex number.................................................... ................... | |||
The plane of complex numbers................................................................... ........................................................ ... | |||
Complex conjugate numbers................................................................... ........................................................ | |||
Operations with complex numbers in algebraic form.................................................... .... | |||
Addition of complex numbers......................................................... ........................................................ | |||
Subtracting complex numbers................................................................... ................................................... | |||
Multiplication of complex numbers................................................................... ............................................... | |||
Dividing complex numbers......................................................... ........................................................ ... | |||
Trigonometric form of writing a complex number.................................................... .......... | |||
Operations with complex numbers in trigonometric form.................................................... | |||
Multiplying complex numbers in trigonometric form.................................................... | |||
Dividing complex numbers in trigonometric form.................................................... ... | |||
Raising a complex number to a positive integer power.................................................... | |||
Extracting the root of a positive integer degree from a complex number.................................... | |||
Raising a complex number to a rational power............................................................ ..... | |||
Complex series................................................... ........................................................ .................... | |||
Complex number series................................................................... ........................................................ | |||
Power series in the complex plane.................................................... ............................ | |||
Two-sided power series in the complex plane.................................................... ... | |||
Functions of a complex variable.................................................... ........................................ | |||
Basic elementary functions......................................................... ......................................... | |||
Euler's formulas................................................... ........................................................ .................... | |||
Exponential form of representing a complex number................................................................. . | |||
Relationship between trigonometric and hyperbolic functions.................................. | |||
Logarithmic function................................................... ........................................................ ... | |||
General exponential and general power functions.................................................... ............... | |||
Differentiation of functions of a complex variable.................................................... ... | |||
Cauchy-Riemann conditions................................................... ........................................................ ............ | |||
Formulas for calculating the derivative.................................................... ................................... | |||
Properties of the differentiation operation................................................................... ........................... | |||
Properties of the real and imaginary parts of an analytical function.................................... | |||
Reconstruction of a function of a complex variable from its real or imaginary |
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Method number 1. Using a curve integral.................................................... ....... | |||
Method No. 2. Direct application of the Cauchy-Riemann conditions.................................... | |||
Method No. 3. Through the derivative of the sought function................................................... ......... | |||
Integration of functions of a complex variable.................................................... .......... | |||
Integral Cauchy formula.................................................... ........................................................ ... | |||
Expansion of functions in Taylor and Laurent series.................................................... ........................... | |||
Zeros and singular points of a function of a complex variable.................................................... ..... | |||
Zeros of a function of a complex variable.................................................... ....................... | |||
Isolated singular points of a function of a complex variable.................................... | |||
14.3 A point at infinity as a singular point of a function of a complex variable
Deductions........................................................ ........................................................ ........................................ | |||
Deduction at the final point................................................... ........................................................ ...... | |||
Residue of a function at a point at infinity.................................................... ............... | |||
Calculation of integrals using residues.................................................... ............................ | |||
Self-test questions................................................................... ........................................................ ....... | |||
Literature................................................. ........................................................ ................................... | |||
Subject index................................................ ........................................................ .............. | |||
Preface
Correctly distributing time and effort when preparing for the theoretical and practical parts of an exam or module certification is quite difficult, especially since there is always not enough time during the session. And as practice shows, not everyone can cope with this. As a result, during the exam, some students solve problems correctly, but find it difficult to answer the simplest theoretical questions, while others can formulate a theorem, but cannot apply it.
These guidelines for preparing for the exam in the course “Theory of Functions of a Complex Variable” (TFCP) are an attempt to resolve this contradiction and ensure simultaneous repetition of the theoretical and practical material of the course. Guided by the principle “Theory without practice is dead, practice without theory is blind,” they contain both theoretical provisions of the course at the level of definitions and formulations, as well as examples illustrating the application of each given theoretical position, and thereby facilitating its memorization and understanding.
The purpose of the proposed methodological recommendations is to help the student prepare for the exam at a basic level. In other words, an extended working guide has been compiled containing the main points used in classes on the TFKP course and necessary when doing homework and preparing for tests. In addition to independent work by students, this electronic educational publication can be used when conducting classes in an interactive form using an electronic board or for placement in a distance learning system.
Please note that this work does not replace either textbooks or lecture notes. For an in-depth study of the material, it is recommended to refer to the relevant sections published by MSTU. N.E. Bauman basic textbook.
At the end of the manual there is a list of recommended literature and a subject index, which includes everything highlighted in the text bold italic terms. The index consists of hyperlinks to sections in which these terms are strictly defined or described and where examples are given to illustrate their use.
The manual is intended for 2nd year students of all faculties of MSTU. N.E. Bauman.
1. Algebraic form of writing a complex number
Notation of the form z = x + iy, where x,y are real numbers, i is an imaginary unit (i.e. i 2 = − 1)
is called the algebraic form of writing a complex number z. In this case, x is called the real part of a complex number and is denoted by Re z (x = Re z), y is called the imaginary part of a complex number and is denoted by Im z (y = Im z).
Example. The complex number z = 4− 3i has a real part Rez = 4 and an imaginary part Imz = − 3.
2. Complex number plane
IN theories of functions of a complex variable are consideredcomplex number plane, which is denoted either by or using letters denoting complex numbers z, w, etc.
The horizontal axis of the complex plane is called real axis, real numbers z = x + 0i = x are placed on it.
The vertical axis of the complex plane is called the imaginary axis;
3. Complex conjugate numbers
The numbers z = x + iy and z = x − iy are called complex conjugate. On the complex plane they correspond to points that are symmetrical about the real axis.
4. Operations with complex numbers in algebraic form
4.1 Addition of complex numbers
The sum of two complex numbers | z 1= x 1+ iy 1 | and z 2 = x 2 + iy 2 is called a complex number |
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z 1+ z 2 | = (x 1+ iy 1) + (x 2+ iy 2) = (x 1+ x 2) + i (y 1+ y 2) . | operation | addition |
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complex numbers is similar to the operation of addition of algebraic binomials. | |||||||||||||
Example. The sum of two complex numbers z 1 = 3+ 7i and z 2 | = −1 +2 i | will be a complex number |
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z 1 +z 2 =(3 +7 i ) +(−1 +2 i ) =(3 −1 ) +(7 +2 ) i =2 +9 i . | |||||||||||||
Obviously, | total amount | conjugate | is | real | |||||||||
z + z = (x+ iy) + (x− iy) = 2 x= 2 Re z. | |||||||||||||
4.2 Subtraction of complex numbers | |||||||||||||
The difference of two complex numbers z 1 = x 1 + iy 1 | X 2 +iy 2 | called | comprehensive |
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number z 1− z 2= (x 1+ iy 1) − (x 2+ iy 2) = (x 1− x 2) + i (y 1− y 2) . | |||||||||||||
Example. The difference of two complex numbers | z 1 =3 −4 i | and z 2 | = −1 +2 i | there will be a comprehensive |
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number z 1 − z 2 = (3− 4i ) − (− 1+ 2i ) = (3− (− 1) ) + (− 4− 2) i = 4− 6i . | |||||||||||||
By difference | complex conjugate | is | |||||||||||
z − z = (x+ iy) − (x− iy) = 2 iy= 2 iIm z. | |||||||||||||
4.3 Multiplication of complex numbers | |||||||||||||
Product of two complex numbers | z 1= x 1+ iy 1 | and z 2= x 2+ iy 2 | called complex |
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z 1z 2= (x 1+ iy 1)(x 2+ iy 2) = x 1x 2+ iy 1x 2+ iy 2x 1+ i 2 y 1y 2 | = (x 1x 2− y 1y 2) + i (y 1x 2+ y 2x) . | ||||||||||||
Thus, the operation of multiplying complex numbers is similar to the operation of multiplying algebraic binomials, taking into account the fact that i 2 = − 1.