Exponential and logarithmic functions of a complex variable. Definition and properties

The main properties of the logarithm, the graph of the logarithm, the domain of definition, the set of values, the basic formulas, increase and decrease are given. Finding the derivative of the logarithm is considered. As well as integral, power series expansion and representation by means of complex numbers.

Content

Range of definition, many values, increasing, decreasing

The logarithm is a monotonic function, therefore it has no extrema. The main properties of the logarithm are presented in the table.

Domain	0 < x < + ∞	0 < x < + ∞
Range of values	- ∞ < y < + ∞	- ∞ < y < + ∞
Monotone	increases monotonically	decreases monotonically
Zeros, y \u003d 0	x \u003d 1	x \u003d 1
Points of intersection with the y-axis, x \u003d 0	no	no
	+ ∞	- ∞
	- ∞	+ ∞

Private values

Logarithm base 10 is called decimal logarithm and denoted as follows:

Logarithm base e called natural logarithm:

Basic formulas for logarithms

Logarithm properties following from the definition of the inverse function:

The main property of logarithms and its consequences

Base replacement formula

Taking the logarithm is a mathematical operation of taking the logarithm. When taking the logarithm, the products of the factors are converted to the sum of the terms.
Potentiation is the inverse mathematical operation of taking logarithms. In potentiation, the given base is raised to the power of the expression over which the potentiation is performed. In this case, the sums of the members are converted into products of factors.

Proof of the main formulas for logarithms

Formulas related to logarithms follow from formulas for exponential functions and from the definition of an inverse function.

Consider the property of the exponential function
.
Then
.
Let's apply the exponential function property
:
.

Let us prove the base change formula.
;
.
Setting c \u003d b, we have:

Inverse function

The inverse of a logarithm to base a is an exponential function with exponent a.

If, then

If, then

Derivative of the logarithm

Derivative of the logarithm of the modulus x:
.
Derivative of the nth order:
.
Derivation of formulas\u003e\u003e\u003e

To find the derivative of the logarithm, it must be reduced to the base e.
;
.

Integral

The integral of the logarithm is calculated by integrating by parts:.
So,

Expressions in terms of complex numbers

Consider the complex number function z:
.
Let us express the complex number z via module r and the argument φ :
.
Then, using the properties of the logarithm, we have:
.
Or

However, the argument φ not uniquely defined. If we put
, where n is an integer,
it will be the same number for different n.

Therefore, the logarithm, as a function of a complex variable, is not an unambiguous function.

Power series expansion

At the decomposition takes place:

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Technical Institutions, "Lan", 2009.

See also:

The exponential function of a real variable (with a positive base) is determined in several steps. First, for natural values \u200b\u200b- as a product of equal factors. The definition is then extended to negative and non-zero integers for by rules. Further, fractional indicators are considered, at which the value of the exponential function is determined using the roots:. For irrational values, the definition is already connected with the basic concept of mathematical analysis - with the passage to the limit, for reasons of continuity. All these considerations are in no way applicable to attempts to extend the exponential function to complex values \u200b\u200bof the indicator, and what is, for example, is completely incomprehensible.

For the first time, a degree with a complex exponent with a natural base was introduced by Euler on the basis of an analysis of a number of constructions of the integral calculus. Sometimes very similar algebraic expressions when integrated give completely different answers:

At the same time, here the second integral is formally obtained from the first when replaced by

Hence, we can conclude that with a proper definition of the exponential function with a complex exponent, inverse trigonometric functions are related to logarithms, and thus the exponential function is related to trigonometric ones.

Euler had the courage and imagination to give a reasonable definition for the exponential function with a basis, namely,

This definition, and therefore this formula is not proven, one can only look for arguments in favor of the rationality and expediency of such a definition. Mathematical analysis provides many arguments of this kind. We will limit ourselves to just one.

It is known that for real there is a limit relation:. On the right-hand side, there is a polynomial that makes sense even for complex values \u200b\u200bfor. The limit of a sequence of complex numbers is determined in a natural way. A sequence is considered to be convergent if sequences of real and imaginary parts converge and is taken

We'll find it. To do this, we turn to the trigonometric form, and for the argument we will choose values \u200b\u200bfrom the interval. With this choice, it is clear that for. Further,

For the passage to the limit, you need to make sure that there are limits for and and find these limits. It is clear that and

So, in the expression

the real part tends to, the imaginary part tends to so that

This simple reasoning provides one of the arguments in favor of Euler's definition of the exponential function.

Let us now establish that when the values \u200b\u200bof the exponential function are multiplied, the indicators add up. Really:

2. Euler's formulas.

We put in the definition of the exponential function. We get:

Replacing b with -b, we get

Adding and subtracting these equalities term by term, we find the formulas

called Euler's formulas. They establish a connection between trigonometric functions and exponential functions with imaginary indicators.

3. Natural logarithm of a complex number.

A complex number given in trigonometric form can be written in the form This form of writing a complex number is called exponential. It retains all the good properties of the trigonometric form, but is even more concise. Further, therefore, it is natural to assume that so that the real part of the logarithm of a complex number is the logarithm of its modulus, the imaginary part is its argument. This to some extent explains the "logarithmic" property of the argument - the argument of the product is equal to the sum of the arguments of the factors.

Plan:

Introduction

• 1 Real logarithm
  • 1.1 Properties
  • 1.2 Logarithmic function
  • 1.3 Natural logarithms
  • 1.4 Decimal logarithms
• 2 Complex logarithm
  • 2.1 Definition and properties
  • 2.2 Examples
  • 2.3 Analytical continuation
  • 2.4 Riemann surface
• 3 Historical sketch
  • 3.1 Real logarithm
  • 3.2 Complex logarithm
• 4 Logarithmic tables
• 5 Applications
Literature
Notes

Introduction

Figure: 1. Graphs of logarithmic functions

Logarithm of a number b by reason a (from the Greek. λόγος - "word", "attitude" and ἀριθμός - "number") is defined as an indicator of the degree to which the base must be built ato get the number b... Designation:. It follows from the definition that the records and are equivalent.

For example, because.

1. Real logarithm

Logarithm of a real number log a b makes sense for. As you know, the exponential function y = a x is monotone and takes each value only once, and the range of its values \u200b\u200bcontains all positive real numbers. It follows that the value of the real logarithm of a positive number always exists and is uniquely determined.

The following types of logarithms are most widely used.

1.1. Properties

Evidence

Let us prove that.

(since by condition bc\u003e 0). ■

Evidence

Let us prove that

(since by condition ■

Evidence

We use identity for the proof. Logarithm both sides of the identity with base c. We get:

Evidence

Let us prove that.

(because b p \u003e 0 by condition). ■

Evidence

Let us prove that

Evidence

Logarithm base left and right c :

Left side: Right side:

Equality of expressions is obvious. Since the logarithms are equal, the expressions themselves are equal due to the monotonicity of the logarithmic function. ■

1.2. Logarithmic function

If we consider the logarithmized number as a variable, we get logarithmic function y \u003d log a x (see fig. 1). It is defined at. Range of values:.

The function is strictly increasing at a \u003e 1 and strictly decreasing at 0< a < 1 . График любой логарифмической функции проходит через точку (1;0) . Функция непрерывна и неограниченно дифференцируема всюду в своей области определения.

Straight x \u003d 0 is the left vertical asymptote, since for a \u003e 1 and at 0< a < 1 .

The derivative of the logarithmic function is:

Evidence

I. Let us prove that

We write the identity e ln x = x and differentiate its left and right parts

We get that, whence it follows that

II. Let us prove that

The logarithmic function implements an isomorphism between the multiplicative group of positive real numbers and the additive group of all real numbers.

1.3. Natural logarithms

Relationship with the decimal logarithm:.

As indicated above, a simple formula is valid for the derivative of the natural logarithm:

For this reason, it is natural logarithms that are mainly used in mathematical research. They often appear when solving differential equations, studying statistical dependencies (for example, the distribution of prime numbers), etc.

The indefinite integral of the natural logarithm is easy to find by integrating by parts:

The Taylor series expansion can be represented as follows:
for, the equality

(1)

In particular,

This series converges faster, and in addition, the left side of the formula can now express the logarithm of any positive number.

1.4. Decimal logarithms

Figure: 2a. Logarithmic scale

Figure: 2b. Logarithmic scale with symbols

Logarithms base 10 (symbol: lg a) before the invention of calculators, they were widely used for calculations. An uneven scale of logarithm decimal is usually applied to slide rulers. A similar scale is used in many areas of science, for example:

• Physics - sound intensity (decibels).
• Astronomy is a scale for the brightness of stars.
• Chemistry is the activity of hydrogen ions (pH).
• Seismology - Richter scale.
• Music theory is a musical scale in relation to the frequencies of musical notes.
• History is a logarithmic time scale.

The logarithmic scale is also widely used to identify the exponent in power dependences and the coefficient in the exponent. In this case, the graph, built on a logarithmic scale along one or two axes, takes the form of a straight line, which is easier to study.

2. Complex logarithm

2.1. Definition and properties

For complex numbers, the logarithm is defined in the same way as real. In practice, almost exclusively the natural complex logarithm is used, which we denote and define as the set of all complex numbers z such that e z = w ... The complex logarithm exists for anyone, and its real part is uniquely determined, while the imaginary one has an infinite set of values. For this reason, it is called a multivalued function. If you imagine w in exemplary form:

,

then the logarithm is found by the formula:

Here is the real logarithm, r = | w | , k - an arbitrary integer. The value obtained when k \u003d 0 is called main value complex natural logarithm; it is customary to take the value of the argument in the interval (- π, π]. The corresponding (already single-valued) function is called main branch logarithm and denoted. Sometimes, through also denotes the value of the logarithm that does not lie on the main branch.

From the formula it follows:

• The real part of the logarithm is determined by the formula:

• The logarithm of a negative number is found by the formula:

Since complex trigonometric functions are related to the exponent (Euler's formula), the complex logarithm, as the inverse to the exponential function, is related to inverse trigonometric functions. An example of such a connection:

2.2. Examples of

Here is the main value of the logarithm for some arguments:

You should be careful when transforming complex logarithms, taking into account that they are multivalued, and therefore the equality of the logarithms of any expressions does not imply equality of these expressions. An example of erroneous reasoning:

iπ \u003d ln (- 1) \u003d ln ((- i) 2) \u003d 2ln (- i) = 2(− iπ / 2) \u003d - iπ - sheer absurdity.

Note that on the left is the main value of the logarithm, and on the right is the value from the lower branch ( k \u003d - 1). The reason for the error is the careless use of the property, which, generally speaking, implies in the complex case the entire infinite set of values \u200b\u200bof the logarithm, and not just the principal value.

2.3. Analytical continuation

Figure: 3. Complex logarithm (imaginary part)

The logarithm of a complex number can also be defined as the analytic continuation of the real logarithm to the entire complex plane. Let the curve Γ begin at unity, does not pass through zero, and does not intersect the negative part of the real axis. Then the principal value of the logarithm at the end point w curve Γ can be determined by the formula:

If Γ is a simple curve (without self-intersections), then for numbers lying on it, logarithmic identities can be used without fear, for example

If the curve Γ is allowed to intersect the negative part of the real axis, then the first such intersection transfers the result from the branch of the principal value to the adjacent branch, and each subsequent intersection causes a similar displacement along the branches of the logarithmic function (see figure).

It follows from the analytic continuation formula that on any branch of the logarithm

For any circle S covering point 0:

The integral is taken in the positive direction (counterclockwise). This identity underlies the theory of residues.

You can also define the analytic continuation of the complex logarithm using the above series (1), generalized to the case of a complex argument. However, from the type of expansion it follows that at unity it is equal to zero, that is, the series refers only to the main branch of the multivalued function of the complex logarithm.

2.4. Riemann surface

A complex logarithmic function is an example of a Riemann surface; its imaginary part (Fig. 3) consists of an infinite number of branches twisted in a spiral. This surface is simply connected; its only zero (first order) is obtained for z \u003d 1, singular points: z \u003d 0 and (branch points of infinite order).

The Riemann surface of the logarithm is a universal covering for the complex plane without point 0.

3. Historical outline

3.1. Real logarithm

The need for complex calculations grew rapidly in the 16th century, and much of the difficulty was associated with multiplying and dividing multidigit numbers, as well as extracting roots. At the end of the century, several mathematicians, almost simultaneously, came up with an idea: to replace time-consuming multiplication with simple addition, comparing geometric and arithmetic progressions with the help of special tables, while the geometric one will be the original one. Then division is automatically replaced by an immeasurably simpler and more reliable subtraction, and the extraction of the root of the power n is reduced to dividing the logarithm of the radical expression by n... He was the first to publish this idea in his book “ Arithmetica integra»Michael Stiefel, who, however, did not make serious efforts to implement his idea.

In 1614, the Scottish amateur mathematician John Napier published an essay in Latin entitled “ Description of the amazing logarithm table"(Lat. Mirifici Logarithmorum Canonis Descriptio ). It contained a short description of logarithms and their properties, as well as 8-digit tables of logarithms of sines, cosines and tangents, with a step of 1 ". logarithm, proposed by Napier, established itself in science. Napier expounded the theory of logarithms in his other book “ Building an amazing logarithm table"(Lat. Mirifici Logarithmorum Canonis Constructio ), published posthumously in 1619 by his son.

The concept of a function did not exist at that time, and Napier defined the logarithm kinematically, comparing uniform and logarithmically slow motion; for example, he defined the logarithm of the sine as follows:

The logarithm of a given sine is a number that always increased arithmetically at the same rate with which the total sine began to decrease geometrically.

In modern notation, Napier's kinematic model can be represented by a differential equation: dx / x \u003d -dy / M, where M is a scale factor introduced so that the value is an integer with the required number of digits (decimal fractions were not yet widely used at that time). Napier took M \u003d 10,000,000.

Strictly speaking, Neper tabulated a different function that is now called the logarithm. If we denote its function LogNap (x), then it is related to the natural logarithm as follows:

Obviously, LogNap (M) \u003d 0, that is, the logarithm of the "full sine" is zero - this is what Napier sought with his definition. ...

The main property of Napier's logarithm: if the quantities form a geometric progression, then their logarithms form an arithmetic progression. However, the rules for taking the logarithm for a neper function were different from those for the modern logarithm.

For instance, LogNap (ab) \u003d LogNap (a) + LogNap (b) - LogNap (1).

Unfortunately, all values \u200b\u200bof Napier's table contained a computational error after the sixth digit. However, this did not prevent the new calculation method from gaining widespread popularity, and many European mathematicians, including Kepler, were engaged in the compilation of logarithmic tables. Already 5 years later, in 1619, the London mathematics teacher John Spidell ( John Speidell) republished Napier's tables, transformed so that they actually became tables of natural logarithms (although Spidell retained the scaling to integers). The term "natural logarithm" was proposed by the Italian mathematician Pietro Mengoli ( Pietro Mengoli)) in the middle of the XVI century.

In the 1620s, Edmund Wingate and William Oughtred invented the first slide rule, an indispensable tool for an engineer before the advent of pocket calculators.

Close to the modern understanding of the logarithm - as an operation inverse to raising to a power - first appeared in Wallis and Johann Bernoulli, and was finally legalized by Euler in the 18th century. In the book "Introduction to the Analysis of Infinite" (1748) Euler gave modern definitions of both exponential and logarithmic functions, led their expansion in power series, especially noted the role of the natural logarithm.

Euler is credited with extending the logarithmic function to a complex domain.

3.2. Complex logarithm

The first attempts to extend logarithms to complex numbers were made at the turn of the 17th-18th centuries by Leibniz and Johann Bernoulli, but they failed to create a holistic theory, primarily for the reason that the very concept of the logarithm was not yet clearly defined. The discussion on this matter was conducted first between Leibniz and Bernoulli, and in the middle of the 18th century - between D'Alembert and Euler. Bernoulli and D'Alembert believed that log (-x) \u003d log (x)... The complete theory of the logarithms of negative and complex numbers was published by Euler in 1747-1751 and is essentially no different from the modern one.

Although the controversy continued (D'Alembert defended his point of view and argued for it in detail in an article in his "Encyclopedia" and in other works), Euler's point of view quickly gained general acceptance.

4. Logarithmic tables

Logarithmic tables

From the properties of the logarithm, it follows that instead of laborious multiplication of multi-digit numbers, it is enough to find (according to tables) and add their logarithms, and then perform potentiation using the same tables, that is, find the value of the result by its logarithm. The division is only different in that the logarithms are subtracted. Laplace said that the invention of logarithms "extended the life of astronomers" by dramatically speeding up the computation process.

When moving a decimal point in a number to n digits the value of the decimal logarithm of this number changes by n ... For example, lg8314.63 \u003d lg8.31463 + 3. Hence it follows that it is enough to compile a table of decimal logarithms for numbers in the range from 1 to 10.

The first tables of logarithms were published by John Napier (1614), and they contained only the logarithms of trigonometric functions, and with errors. Yost Burgi, a friend of Kepler's (1620), published his tables independently. In 1617, Oxford professor of mathematics, Henry Briggs, published tables that already included the decimal logarithms of the numbers themselves, from 1 to 1000, with 8 (later 14) digits. But Briggs' tables also showed errors. The first unmistakable edition based on Vega tables (1783) did not appear until 1857 in Berlin (Bremiver tables).

In Russia, the first tables of logarithms were published in 1703 with the participation of L. F. Magnitsky. Several collections of tables of logarithms were published in the USSR.

• Bradis V.M. Four-digit math tables. 44th edition, M., 1973.

Bradis tables (1921) were used in educational institutions and in engineering calculations that do not require great accuracy. They contained the mantissa of decimal logarithms of numbers and trigonometric functions, natural logarithms, and some other useful calculation tools.

• Vega G. Tables of seven-digit logarithms, 4th edition, M., 1971.

Professional collection for accurate calculations.

• Five-digit tables of natural values \u200b\u200bof trigonometric quantities, their logarithms and logarithms of numbers, 6th ed., Moscow: Nauka, 1972.
• Tables of natural logarithms, 2nd edition, in 2 volumes, Moscow: Nauka, 1971.

Nowadays, with the spread of calculators, the need to use tables of logarithms has disappeared.

M, Feature (complex analysis).

Proof of the formula .

=

= =

since the sine and cosine do not depend on the addition of an angle multiple

And this equality is already obvious, since this is the trigonometric form of a complex number.

Thus, the logarithm exists for all points in the plane except zero. For a real positive number, the argument is 0, so this infinite set of points has the form, that is, one of the values, namely, at, will fall on the real axis. If we calculate the logarithm of a negative number, then we get , that is, the set of points is shifted upward and none of them fall on the actual axis.

It can be seen from the formula that only with a zero argument of the initial number, one of the values \u200b\u200bof the logarithm falls on the real axis. And this corresponds to the right semiaxis, and that is why only logarithms of positive numbers were considered in the school mathematics course. Logarithms of negative and imaginary numbers also exist, but they do not have a single value on the real axis.

The following drawing shows where all the values \u200b\u200bof the logarithm of a positive number are located in the plane. One of them is on the real axis, the rest are higher and lower on,, and so on. For a negative or complex number, the argument is nonzero, so this sequence of points vertically shifts, resulting in no point on the real axis.

Example. Calculate.

Decision. Let's define the modulus of the number (equal to 2) and the argument 180 0, that is. Then \u003d .

Appendix 1. Questions for evidence (for tickets).

Lecture number 1

1. Prove the formula for integration by parts.

Lecture number 2

1. Prove that the change, where r \u003d LCM (r 1, ..., r k) reduces the integral

2. Prove that a change of change reduces an integral of the form to the integral of the rational fraction.

3. Derive transformation formulas for sine and cosine

For generic trigonometric substitution.

4. Prove that in the case when the function is odd with respect to the cosine, the change reduces the integral to a rational fraction.

5. Prove that in the case when

replacement: reduces the integral to a rational fraction.

6. Prove that for an integral of the form

7. Prove the formula

8. Prove that for an integral of the form the change reduces the integral to a rational fraction.

9. Prove that for an integral of the form the replacement reduces the integral to a rational fraction.

Lecture number 3

1. Prove that the function is the antiderivative of a function.

2. Prove the Newton-Leibniz formula: .

3. Prove the formula for the length of an explicitly given curve:

.

4. Prove the formula for the length of a curve given in polar coordinates

Lecture number 4

Prove the theorem: converges, converges.

Lecture number 5

1. Derive (prove) the formula for the area of \u200b\u200ban explicitly given surface .

2. Derivation of formulas for the transition to polar coordinates.

3. Derivation of the Jacobi determinant of polar coordinates.

4. Derivation of formulas for the transition to cylindrical coordinates.

5. Derivation of the Jacobi determinant of cylindrical coordinates.

6. Derivation of the formulas for the transition to spherical coordinates:

.

Lecture number 6

1. Prove that the change reduces the homogeneous equation to an equation with separable variables.

2. Derive the general form of the solution to a linear homogeneous equation.

3. Derive the general form of the solution of a linear inhomogeneous equation by the Lagrange method.

4. Prove that the change reduces the Bernoulli equation to a linear equation.

Lecture number 7.

1. Prove that the change lowers the order of the equation by k.

2. Prove that the change lowers by one the order of the equation .

3. Prove the theorem: The function is a solution to a linear homogeneous differential equation is the characteristic root.

4. Prove the theorem that a linear combination of solutions to a linear homogeneous diff. equations are also its solution.

5. Prove the theorem on the superposition of solutions: If is a solution of a linear inhomogeneous differential equation with the right-hand side, and is a solution of the same differential equation, but with the right-hand side, then the sum is a solution to the equation with the right-hand side.

Lecture number 8.

1. Prove the theorem that the system of functions is linearly dependent.

2. Prove the theorem that there are n linearly independent solutions of a linear homogeneous differential equation of order n.

3. Prove that if 0 is a root of multiplicity, then the system of solutions corresponding to this root has the form.

Lecture number 9.

1. Prove with the help of the exponential form that when multiplying complex numbers, the modules are multiplied, and the arguments are added.

2. Prove the Moivre formula for degree n

3. Prove the formula for the root of order n of a complex number

.

4. Prove that and

are generalizations of sine and cosine, i.e. for real numbers, these formulas will give a sine (cosine).

5. Prove the formula for the logarithm of a complex number:

Appendix 2.

Minor and oral questions for knowledge of theory (for colloquia).

Lecture number 1

1. What are antiderivatives and indefinite integrals, how do they differ?

2. Explain why it is also antiderivative.

3. Write the formula for integration by parts.

4. What replacement is required in an integral of the form and how does it eliminate roots?

5. Write down the form of the decomposition of the integrand of the rational fraction into the elementary ones in the case when all the roots are different and real.

6. Write down the form of the decomposition of the integrand of the rational fraction into elementary ones in the case when all the roots are real and there is one multiple root of multiplicity k.

Lecture number 2.

1. Write what is the decomposition of a rational fraction into elementary ones in the case when the denominator has a factor of 2 degrees with a negative discriminant.

2. What replacement reduces the integral to a rational fraction?

3. What is universal trigonometric substitution?

4. What changes are made in cases when the function under the integral sign is odd with respect to the sine (cosine)?

5. What replacements are made in the case of the presence of expressions,, or in the integrand.

Lecture number 3.

1. Definition of a definite integral.

2. List some of the basic properties of a definite integral.

3. Write the Newton-Leibniz formula.

4. Write the formula for the volume of the body of revolution.

5. Write the formula for the length of an explicitly defined curve.

6. Write the formula for the length of a parametrically defined curve.

Lecture number 4.

1. Determination of improper integral (using the limit).

2. What is the difference between improper integrals of the 1st and 2nd kind.

3. Give simple examples of convergent integrals of the 1st and 2nd kind.

4. For which integrals (T1) converge.

5. How convergence is related to the finite limit of the antiderivative (T2)

6. What is a necessary convergence criterion, its formulation.

7. Sign of comparison in the final form

8. Sign of comparison in the limiting form.

9. Definition of multiple integrals.

Lecture number 5.

1. Change of the order of integration, to show on the simplest example.

2. Write the surface area formula.

3. What are polar coordinates, write the transition formulas.

4. What is the Jacobian of the polar coordinate system?

5. What are cylindrical and spherical coordinates, what is their difference.

6. What is the Jacobian of cylindrical (spherical) coordinates?

Lecture number 6.

1. What is a 1st order differential equation (general view).

2. What is a differential equation of the 1st order, solved with respect to the derivative. Give an example.

3. What is an equation with separable variables.

4. What is the general, particular solution, the Cauchy conditions.

5. What is a homogeneous equation, what is the general method for solving it.

6. What is a linear equation, what is the algorithm for solving it, what is the Lagrange method.

7. What is the Bernoulli equation, the algorithm for its solution.

Lecture number 7.

1. What replacement is needed for an equation of the form.

2. What replacement is needed for an equation of the form .

3. Show with examples how you can express as.

4. What is a linear differential equation of order n.

5. What is a characteristic polynomial, characteristic equation.

6. Formulate a theorem on for which r the function is a solution to a linear homogeneous differential equation.

7. Formulate a theorem that a linear combination of solutions to a linear homogeneous equation is also its solution.

8. Formulate a theorem on the superposition of solutions and its consequences.

9. What are linearly dependent and linearly independent systems of functions, give some examples.

10. What is the Wronsky determinant of a system of n functions, give an example of the Wronsky determinant for LZS and LNS systems.

Lecture number 8.

1. What property does the Wronski determinant have if the system is a function linearly dependent.

2. How many linearly independent solutions of a linear homogeneous differential equation of order n exist.

3. Determination of the FSR (fundamental system of solutions) of a linear homogeneous equation of order n.

4. How many functions are there in the SDF?

5. Write down the form of the system of equations to be found by the Lagrange method for n \u003d 2.

6. Write down the type of private decision in the case when

7. What is a linear system of differential equations, write some example.

8. What is an autonomous system of differential equations.

9. The physical meaning of the system of differential equations.

10. Write down what functions the FSR of a system of equations consists of if the eigenvalues \u200b\u200band eigenvectors of the main matrix of this system are known.

Lecture number 9.

1. What is an imaginary unit.

2. What is a conjugate number and what happens when you multiply it by the original.

3. What is the trigonometric, exponential form of a complex number.

4. Write down Euler's formula.

5. What is a module, an argument of a complex number.

6. what happens to modules and arguments during multiplication (division).

7. Write the Moivre formula for the degree n.

8. Write the formula for the root of order n.

9. Write the generalized sine and cosine formulas for the complex argument.

10. Write the formula for the logarithm of a complex number.

Appendix 3. Tasks from the lectures.

Lecture number 1

Example. . Example. .

Example. . Example. .

Example. Example. .

Example. . Example. .

Lecture number 2

Example. . Example. .

Example. . Example. .

Example. . Example. ... , where, number.

Example. Divide in exemplary form.

Example... Find by the Moivre formula.

Example... Find all root values.