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 in the fact that new numbers (imaginary) are added to the real numbers. The introduction of these numbers is connected with the impossibility in the set of real numbers of extracting the root from a negative number.
Introduction of the concept of a complex number.
The imaginary numbers with which we supplement the real numbers are written as bi, where i is the 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 are 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, b1=b2.
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 is a 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 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, which differ only in the sign of the imaginary part, are called conjugate.
Actions on complex numbers in algebraic form.
The following operations can be performed on complex numbers in algebraic form.
1) Addition.
Definition. The sum of complex numbers z 1 = a 1 + b 1 i and z 2 = a 2 + b 2 i called a complex number z, the real part of which is equal to the sum of the real parts z1 and z2, and the imaginary part is the sum of the imaginary parts of the numbers z1 and z2, i.e z = (a 1 + a 2) + (b 1 + b 2)i.
Numbers z1 and z2 are called terms.
The addition of complex numbers has the following properties:
1º. Commutativity: z1 + z2 = z2 + z1.
2º. Associativity: (z 1 + z 2) + z 3 = z 1 + (z 2 + z 3).
3º. Complex number -a -bi is 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 equals zero: z + (-z) = 0
Example 1: Add (3 - i) + (-1 + 2i).
(3 - i) + (-1 + 2i) = (3 + (-1)) + (-1 + 2) i = 2 + 1i.
2) Subtraction.
Definition. Subtract from complex number z1 complex number z2 z, what z + z 2 = z 1.
Theorem. The difference of complex numbers exists and, moreover, is unique.
Example 2: Subtract (4 - 2i) - (-3 + 2i).
(4 - 2i) - (-3 + 2i) = (4 - (-3)) + (-2 - 2) i = 7 - 4i.
3) Multiplication.
Definition. The product of complex numbers z 1 =a 1 +b 1 i and z 2 \u003d a 2 + b 2 i 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 z1 and z2 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 with respect to addition:
(z 1 + z 2) z 3 \u003d z 1 z 3 + z 2 z 3.
4º. z \u003d (a + bi) (a - bi) \u003d a 2 + b 2 is a real number.
In practice, the multiplication of complex numbers is carried out according to the rule of multiplying the sum by the sum and separating the real and imaginary parts.
In the following example, consider the multiplication of complex numbers in two ways: by the rule and by multiplying the sum by the sum.
Example 3: Multiply (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.
2 way. (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 z1 to a complex number z2, 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 z2 ≠ 0 + 0i.
In practice, the quotient of complex numbers is found by multiplying the numerator and denominator by the conjugate of the denominator.
Let be z 1 = a 1 + b 1 i, z 2 = a 2 + b 2 i, then
.
In the following example, we perform division by the formula and the rule of multiplication by the conjugate of the denominator.
Example 4. Find a quotient 
.
5) Raising to a positive integer power.
a) Powers of the imaginary unity.
Taking advantage of the equality i 2 \u003d -1, it is easy to define any positive integer power of the imaginary unit. We have:
i 3 \u003d i 2 i \u003d -i,
i 4 \u003d i 2 i 2 \u003d 1,
i 5 \u003d i 4 i \u003d i,
i 6 \u003d i 4 i 2 \u003d -1,
i 7 \u003d i 5 i 2 \u003d -i,
i 8 = i 6 i 2 = 1 etc.
This shows that the degree values i n, where n- a positive integer, periodically repeated when the indicator increases by 4 .
Therefore, to raise the number i to a positive integer power, divide the exponent by 4 and erect i to the power whose exponent is 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 \u003d (1 + i) (- i) \u003d - i + 1 \u003d 1 - i.
b) Raising a complex number to a positive integer power is carried out according to the rule of 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.
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Algebraic form of a complex number.
Addition, subtraction, multiplication and division of complex numbers.
We have already met 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 differ little from ordinary algebra.
Addition of complex numbers
Example 1
Add two complex numbers,
To add two complex numbers, add their real and imaginary parts:
Simple, isn't it? The action is so obvious that it does not need additional comments.
In such a 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 true: 
- from the rearrangement of the terms, the sum does not change.
Subtraction of complex numbers
Example 2
Find the differences of complex numbers and , if ,
The action is similar to addition, the only feature is that the subtrahend must be taken in brackets, and then, as a standard, open these brackets with a sign change:
The result should not confuse, the resulting number has two, not three parts. Just the real part is a component: . For clarity, the answer can be rewritten as follows: .
Let's calculate the second difference:
Here the real part is also a component:
To avoid any understatement, I will give a short example with a "bad" imaginary part: . Here you can't do without parentheses.
Multiplication of complex numbers
The moment 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 is being asked? It suggests itself to open the brackets according to the rule of multiplication of polynomials. That's how it should be done! All algebraic operations are familiar to you, the main thing to remember is that and be careful.
Let's 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 the other polynomial.
I will write in detail:
I hope it was clear to everyone that
Attention, and again attention, most often a mistake is made in signs.
Like the sum, the product of complex numbers is permutable, that is, the equality is true: .
In educational literature and on the Web, 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 multiplication of polynomials is more universal and clearer. I will not give the formula, I think that in this case it is clogging the head with sawdust.
Division of complex numbers
Example 4
Given complex numbers , . Find private.
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.
We recall 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, multiply the numerator by the same number:
I will write in detail:
I picked up a “good” example, if you take two numbers “from the bulldozer”, then as a result of division you will almost always get fractions, something like.
In some cases, before dividing, it is advisable to simplify the fraction, 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, I will give the correct answer:
Rarely, but there is such a task:
Example 5
You are given a complex number. Write the given number in algebraic form (i.e. in the form).
The reception 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 has , so the denominator and numerator must be multiplied by the conjugate expression, that is, by:
In practice, they can easily offer a fancy example where you need to perform a lot of operations with complex numbers. No Panic: be careful, follow the rules of algebra, the usual algebraic order of operations, and remember that .
Trigonometric and exponential form of a complex number
In this section, we will focus more on the trigonometric form of a complex number. The exponential form in practical tasks is much less common. 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 complex number modulus, a - complex number argument. Don't run away, it's easier than you think.
Draw a number on the complex plane. For definiteness and simplicity of explanations, we will place it in the first coordinate quarter, i.e. we think that:
The modulus of a complex number is the distance from the origin of coordinates to the corresponding point of the complex plane. Simply put, modulus is the length radius vector, which is marked 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 valid for any meanings "a" and "be".
Note: the modulus of a complex number is a generalization of the concept real number modulus, as the distance from the point to the origin.
The argument of a complex number called injection between positive 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 under consideration is actually similar to polar coordinates, where the polar radius and polar angle uniquely define a point.
The argument of a complex number is usually denoted by: or
From geometric considerations, the following formula for finding the argument is obtained:
. Attention! This formula works only 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 consider these cases.
But first, consider the simplest examples, when complex numbers are located on the coordinate axes.
Example 7
Let's execute the drawing:
In fact, the task is oral. For clarity, I will rewrite the trigonometric form of a complex number: 
Let's remember tightly, the module - length(which is always non-negative ), the argument is injection.
1) Let's represent the number in trigonometric form. Find its modulus and argument. It's obvious that . Formal calculation according to the formula: .
It is obvious that (the number lies directly on the real positive semiaxis). So the number in trigonometric form is: 
.
Clear as day, reverse check action:
2) Let's represent the number in trigonometric form. Find its modulus and argument. It's obvious that . Formal calculation according to the formula: .
Obviously (or 90 degrees). In the drawing, the corner is marked 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 a number (at the same time by checking):
3) Let's represent the number in trigonometric form. Find its modulus and argument. It's obvious that . Formal calculation according to the formula: .
Obviously (or 180 degrees). In the drawing, the angle 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. Find its modulus and argument. It's obvious that . Formal calculation according to 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 becomes: 
Attention! In no case should you use the evenness of the cosine, the oddness of the sine and carry out further "simplification" of the record:
By the way, it is useful to recall 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 are much easier to learn!
In the design of the simplest examples, it should be written like this: “it is obvious that the module is ... it is obvious that the argument is ...”. This is really obvious and easily solved verbally.
Let's move on to 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 rewrite them in your notebook):
1) If (1st and 4th coordinate quarters, or the right half-plane), then the argument must be found using the formula.
2) If (2nd coordinate quarter), then the argument must be found by the formula 
.
3) If (3rd coordinate quarter), then the argument must be found by the formula 
.
Example 8
Express the complex numbers in trigonometric form: , , , .
As soon as there are ready-made formulas, then the drawing is not necessary. But there is one point: when you are asked to present a number in trigonometric form, then drawing is better to do anyway. The fact is that teachers often reject a solution without a drawing, the absence of a drawing is a serious reason for a minus and a failure.
Eh, I haven’t drawn anything by hand for a hundred years, hold on:
As always, messy turned out =)
I will present the numbers and in complex form, the first and third numbers will be for independent decision.
Let's represent the number in trigonometric form. Find its modulus and argument.
Algebraic form of writing a complex number .............................................. ................... | |||
Plane of complex numbers .................................................................. ................................................. ... | |||
Complex conjugate numbers ............................................................... ................................................. | |||
Operations with complex numbers in algebraic form .............................................................. .... | |||
Addition of complex numbers .................................................................. ................................................. | |||
Subtraction of complex numbers .......................................................... ................................................ | |||
Multiplication of complex numbers .......................................................... ............................................... | |||
Division of complex numbers .............................................................. ................................................. ... | |||
Trigonometric form of a complex number .............................................................. .......... | |||
Operations with complex numbers in trigonometric form ....................................................... | |||
Multiplication of complex numbers in trigonometric form.................................................................... | |||
Division of complex numbers in trigonometric form .............................................................. ... | |||
Raising a complex number to a positive integer power | |||
Extracting the root of a positive integer power 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 formulas .................................................. ................................................. .................... | |||
The exponential form of the representation of 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 operation of differentiation .......................................................... ............................... | |||
Properties of the real and imaginary parts of an analytic function .............................................. | |||
Recovery of a function of a complex variable from its real or imaginary |
|||
Method number 1. Using the Curvilinear Integral ....................................................... ....... | |||
Method number 2. Direct application of the Cauchy-Riemann conditions....................................... | |||
Method number 3. Through the derivative of the desired function .............................................................. ......... | |||
Integration of functions of a complex variable............................................................... ........... | |||
Integral formula of Cauchy .............................................. ................................................. ... | |||
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 Point at infinity as a singular point of a function of a complex variable
Withdrawals ................................................. ................................................. ................................................ | |||
Deduction at the end point .......................................................... ................................................. ...... | |||
Residue of a function at a point at infinity .............................................................. ................. | |||
Calculation of integrals using residues .............................................................. ............................... | |||
Questions for self-examination ............................................................... ................................................. ....... | |||
Literature................................................. ................................................. ................................. | |||
Subject index................................................ ................................................. .............. | |||
Foreword
It is quite difficult to correctly allocate time and effort in preparing for the theoretical and practical parts of an exam or module certification, 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 correctly solve problems, but find it difficult to answer the simplest theoretical questions, while others can formulate a theorem, but cannot apply it.
These methodological recommendations for preparing for the exam in the Theory of Functions of a Complex Variable (TFV) course 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 the theoretical positions of the course at the level of definitions and formulations, and 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 that contains the main points used in the TFKT course classes and necessary when doing homework and preparing for control activities. In addition to independent work of 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 textbooks or lecture notes. For an in-depth study of the material, it is recommended to refer to the relevant sections of the publication published at the Moscow State Technical University. N.E. Bauman basic textbook.
At the end of the manual there is a list of recommended literature and a subject index, which includes all those highlighted in the text. bold italic terms. The index consists of hyperlinks to sections where 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
Recording of the form z \u003d x + iy, where x, y are real numbers, i is an imaginary unit (i.e. i 2 = − 1)
is called the algebraic form of the complex number z. In this case, x is called the real part of the complex number and denoted by Re z (x = Re z ), y is called the imaginary part of the complex number and denoted by Im z (y = Im z ).
Example. The complex number z = 4− 3i has the real part Rez = 4 , and the imaginary part Imz = − 3 .
2. Plane of complex numbers
AT theories of functions of a complex variable considercomplex number plane, which is denoted either, or the letters denoting complex numbers z, w, etc. are used.
The horizontal axis of the complex plane is called real axis, real numbers are located on it z \u003d x + 0i \u003d x.
The vertical axis of the complex plane is called the imaginary axis, it has
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 | additions |
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complex numbers is similar to the operation of adding 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, | sum in a complex | conjugated | is an | valid | |||||||||
z + z = (x + iy) + (x − iy) = 2 x= 2 Rez. | |||||||||||||
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 between two complex numbers | z 1 =3 −4 i | and z2 | = −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 . | |||||||||||||
difference | complex conjugate | is an | |||||||||||
z − z = (x+ iy) − (x − iy) = 2 iy= 2 iIm z. | |||||||||||||
4.3 Multiplication of complex numbers | |||||||||||||
The product of two complex numbers | z 1= x 1+ iy 1 | and z 2= x 2+ iy 2 | is 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 multiplication of complex numbers is similar to the operation of multiplication of algebraic binomials, taking into account the fact that i 2 = − 1.
