Laplace transforms examples. Laplace transform. To solve linear differential equations we will use the Laplace transform. The essence of the Laplace transform

This is the name of another type of integral transformation, which, along with the Fourier transform, is widely used in radio engineering to solve a wide variety of problems related to the study of signals.

The concept of complex frequency.

Spectral methods, as is already known, are based on the fact that the signal under study is represented as the sum of an unlimited number of elementary terms, each of which periodically changes in time according to the law.

A natural generalization of this principle lies in the fact that instead of complex exponential signals with purely imaginary exponents, exponential signals of the form are introduced into consideration, where is a complex number: called the complex frequency.

From two such complex signals, a real signal can be composed, for example, according to the following rule:

where is the complex conjugate quantity.

Indeed, at the same time

Depending on the choice of the real and imaginary parts of the complex frequency, a variety of real signals can be obtained. So, if , but we get ordinary harmonic oscillations of the form If then, depending on the sign, we get either increasing or decreasing exponential oscillations in time. Such signals acquire a more complex form when. Here the multiplier describes an envelope that changes exponentially over time. Some typical signals are shown in Fig. 2.10.

The concept of complex frequency turns out to be very useful, primarily because it makes it possible, without resorting to generalized functions, to obtain spectral representations of signals, mathematical models which are non-integrable.

Rice. 2.10. Real signals corresponding to different values ​​of complex frequency

Another consideration is also significant: exponential signals of the form (2.53) serve as a “natural” means of studying oscillations in various linear systems. These issues will be explored in Chap. 8.

It should be noted that the true physical frequency co serves as the imaginary part of the complex frequency. There is no special term for the real part of the complex frequency.

Basic relationships.

Let be some signal, real or complex, defined at t > 0 and equal to zero at negative times. The Laplace transform of this signal is a function of a complex variable given by the integral:

The signal is called the original, and the function is called its Laplace image (for short, just an image).

The condition that ensures the existence of integral (2.54) is as follows: the signal must have no more than an exponential degree of growth, i.e., it must satisfy the inequality where are positive numbers.

When this inequality is satisfied, the function exists in the sense that the integral (2.54) converges absolutely for all complex numbers for which Number a is called the abscissa of absolute convergence.

The variable in the basic formula (2.54) can be identified with the complex frequency. Indeed, at a purely imaginary complex frequency, when formula (2.54) turns into formula (2.16), which determines the Fourier transform of the signal, which is equal to zero at Thus, the Laplace transform can be considered

Just as this is done in the theory of the Fourier transform, it is possible, knowing the image, to restore the original. To do this, in the formula for the inverse Fourier transform

it is necessary to carry out an analytical continuation, moving from the imaginary variable to the complex argument a On the complex frequency plane, integration is carried out along an unlimitedly extended vertical axis located to the right of the abscissa of absolute convergence. Since at differential , the formula for the inverse Laplace transform takes the form

In the theory of functions of a complex variable, it has been proven that Laplace images have “good” properties in terms of smoothness: such images at all points of the complex plane, with the exception of a countable set of so-called singular points, are analytic functions. Singular points, as a rule, are poles, single or multiple. Therefore, to calculate integrals of the form (2.55), flexible methods of residue theory can be used.

In practice, Laplace transform tables are widely used, which collect information about the correspondence between the originals. and images. The presence of tables has made the Laplace transform method popular both in theoretical research and in engineering calculations of radio engineering devices and systems. In the Appendices there is such a table that allows you to solve a fairly wide range of problems.

Examples of calculating Laplace transforms.

The way images are calculated has a lot in common with what has already been studied in relation to the Fourier transform. Let's consider the most typical cases.

Example 2.4, Illustration of generalized exponential momentum.

Let , where is a fixed complex number. The presence of the -function determines the equality at Using formula (2.54), we have

If then the numerator goes to zero when the upper limit is substituted. As a result, we obtain the correspondence

As a special case of formula (2.56), we can find an image of a real exponential video pulse:

and complex exponential signal:

Finally, putting in (2.57) we find the image of the Heaviside function:

Example 2.5. Illustration of a delta function.

Laplace transform- integral transformation connecting the function F(s) (\displaystyle \F(s)) complex variable ( image) with function f (x) (\displaystyle \f(x)) real variable ( original). With its help, the properties of dynamic systems are studied and differential and integral equations are solved.

One of the features of the Laplace transform, which predetermined its wide distribution in scientific and engineering calculations, is that many relations and operations on the originals correspond to simpler relations on their images. Thus, the convolution of two functions is reduced in image space to a multiplication operation, and linear differential equations become algebraic.

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Subtitles

Definition

Direct Laplace transform

f(x) | e − σ 0 x d x = ∫ 0 ∞ | (f(x) | e − σ 0 x d x , (\displaystyle \lim _(b\to \infty )\int \limits _(0)^(b)|f(x)|e^(-\sigma _(0)x)\ ,dx=\int \limits _(0)^(\infty )|f(x)|e^(-\sigma _(0)x)\,dx,) then it converges absolutely and uniformly for and is an analytical function atσ ⩾ σ 0 (\displaystyle \sigma \geqslant \sigma _(0)) σ = R e s (\displaystyle \sigma =\mathrm (Re) \,s)- real part of a complex variable s (\displaystyle s)). Exact bottom edge σ a (\displaystyle \sigma _(a)) sets of numbers

• Conditions for the existence of the direct Laplace transform

Laplace transform L ( f (x) ) (\displaystyle (\mathcal (L))\(f(x)\)) exists in the sense of absolute convergence in following cases:

1. σ ⩾ 0 (\displaystyle \sigma \geqslant 0): Laplace transform exists if integral exists ∫ 0 ∞ |;
2. f(x) | d x (\displaystyle \int \limits _(0)^(\infty )|f(x)|\,dx) σ > σ a (\displaystyle \sigma >\sigma _(a)): Laplace transform exists if the integral ∫ 0 x 1 | f(x) | d x (\displaystyle \int \limits _(0)^(x_(1))|f(x)|\,dx) exists for every finite x 1 > 0 (\displaystyle x_(1)>0);
3. And| f(x) | f(x) | ⩽ K e σ a x (\displaystyle |f(x)|\leqslant Ke^(\sigma _(a)x)) For x > x 2 ⩾ 0 (\displaystyle x>x_(2)\geqslant 0)σ > 0 (\displaystyle \sigma >0) f(x) |.

or

• (which bound is greater): a Laplace transform exists if a Laplace transform exists for the function

f ′ (x) (\displaystyle f"(x))

1. (derivative of f (x) (\displaystyle f(x))) For Note Conditions for the existence of the inverse Laplace transform For the existence of the inverse Laplace transform, it is sufficient to satisfy the following conditions: exists for every finite If the image.
2. F (s) (\displaystyle F(s)) - analytical function forσ ⩾ σ a (\displaystyle \sigma \geqslant \sigma _(a)) and has an order less than −1, then the inverse transformation for it exists and is continuous for all values ​​of the argument, and L − 1 ( F (s) ) = 0 (\displaystyle (\mathcal (L))^(-1)\(F(s)\)=0) t ⩽ 0 (\displaystyle t\leqslant 0) Let F (s) = φ [ F 1 (s) , F 2 (s) , … , F n (s) ] (\displaystyle F(s)=\varphi ), So φ (z 1 , z 2 , … , z n) (\displaystyle \varphi (z_(1),\;z_(2),\;\ldots ,\;z_(n))) analytical regarding each

or z k (\displaystyle z_(k))

• and is equal to zero for

z 1 = z 2 = … = z n = 0 (\displaystyle z_(1)=z_(2)=\ldots =z_(n)=0) and is equal to zero for

• , And

F k (s) = L ( f k (x) ) (σ > σ a k: k = 1 , 2 , … , n) (\displaystyle F_(k)(s)=(\mathcal (L))\(f_ (k)(x)\)\;\;(\sigma >\sigma _(ak)\colon k=1,\;2,\;\ldots ,\;n))

(\displaystyle (\mathcal (L))\(f"(x)\)=s\cdot F(s)-f(0^(+)).)

are in the left half-plane.

• The Finite Value Theorem is very useful because it describes the behavior of the original at infinity using a simple relation. This is, for example, used to analyze the stability of the trajectory of a dynamic system.

Other properties

L ( a f (x) + b g (x) ) = a F (s) + b G (s) .

Multiplying by a number:

L ( f (a x) ) = 1 a F (s a) .

Section II. Mathematical analysis

E. Yu. Anokhina

HISTORY OF DEVELOPMENT AND ESTABLISHMENT OF THE THEORY OF THE FUNCTION OF A COMPLEX VARIABLE (TFCV) AS A EDUCATIONAL SUBJECT

One of the complex mathematical courses is the TFKP course. The complexity of this course is due, first of all, to the variety of its relationships with other mathematical disciplines, historically expressed in the broad applied focus of the science of TFKP.

In the scientific literature on the history of mathematics, there is scattered information about the history of the development of TFKP; they require systematization and generalization.

In this regard, the main objective of this article is short description development of TFKP and the formation of this theory as an educational subject.

As a result of the study, the following three stages in the development of TFKP as a science and educational subject were identified:

The stage of emergence and recognition of complex numbers;

The stage of accumulation of factual material on functions of imaginary quantities;

The stage of formation of the theory of functions of a complex variable.

The first stage of the development of TFKP (mid-16th century - 18th century) begins with the work of G. Cardano (1545) who published the work “Artis magnae sive de regulis algebraitis” (Great art, or on algebraic rules). The main task of G. Cardano's work was to justify general algebraic techniques for solving equations of the third and fourth degrees, recently discovered by Ferro (1465-1526), ​​Tartaglia (1506-1559) and Ferrari (1522-1565). If the cubic equation is reduced to the form

x3 + px + d = 0,

and it should be

When (t^Ar V (|- 70 the equation has three real roots, and two of them

are equal to each other. If then the equation has one real and two co-

conjugate complex roots. Complex numbers appear in the final result, so G. Cardano could do what they did before: declare the equation to have

one root. When (<7 Г + (р V < (). тогда уравнение имеет три действительных корня. Этот так

the so-called irreducible case is characterized by one feature that was not encountered until the 16th century. The equation x3 - 21x + 20 = 0 has three real roots 1, 4, - 5, which is easy

verify by simple substitution. But ^du + y _ ^20y + ^-21y _ ^ ^ ^; therefore, according to the general formula, x = ^-10 + ^-243 -^-10-4^243. Complex, i.e. “false”, the number turns out to be not a result, but an intermediate term in the calculations that lead to the real roots of the equation in question. J. Cardano encountered a difficulty and realized that in order to preserve the generality of this formula, it was necessary to abandon the complete disregard of complex numbers. J. d'Alembert (1717-1783) believed that it was precisely this circumstance that made G. Cardano and mathematicians following this idea seriously interested in complex numbers.

At this stage (in the 17th century) two points of view were generally accepted. The first point of view was expressed by Girard, who raised the question of recognizing the need for the unrestricted use of complex numbers. The second is by Descartes, who denied the possibility of interpreting complex numbers. Opposite to Descartes's opinion was the point of view of J. Wallis - about the existence of a real interpretation of complex numbers, Descartes ignored it. Complex numbers began to be “forced” to be used in solving applied problems in situations where the use of real numbers led to a complex result, or the result could not be obtained theoretically, but had a practical implementation.

The intuitive use of complex numbers led to the need to preserve the laws and rules of arithmetic of real numbers for the set of complex numbers; in particular, there were attempts at direct transfer. This sometimes led to erroneous results. In this regard, questions about the justification of complex numbers and the construction of algorithms for their arithmetic have become relevant. This was the beginning of a new stage in the development of TFKP.

The second stage of development of TFKP (beginning of the 18th century - 19th century). In the 18th century L. Euler expressed the idea that the field of complex numbers is algebraically closed. The algebraic closedness of the field of complex numbers C led mathematicians to the following conclusions:

That the study of functions and mathematical analysis in general acquire due completeness and completeness only when considering the behavior of functions in a complex domain;

It is necessary to consider complex numbers as variables.

In 1748, L. Euler (1707-1783) in his work “Introduction to the Analysis of Infinitesimals” introduced a complex variable as the most general concept of a variable quantity, using complex numbers in the expansion of functions into linear factors. L. Euler is rightfully considered one of the creators of TFKP. In the works of L. Euler, elementary functions of a complex variable were studied in detail (1740-1749), conditions for differentiability were given (1755) and the beginning of integral calculus for functions of a complex variable (1777). L. Euler practically introduced the conformal mapping (1777). He called these mappings “similar in the small,” and the term “conformal” was apparently first used by the St. Petersburg academician F. Schubert (1789). L. Euler also brought numerous applications of functions of a complex variable to various mathematical problems and laid the foundation for their use in hydrodynamics (1755-1757) and cartography (1777). K. Gauss formulates the definition of an integral in the complex plane, an integral theorem on the decomposability of an analytic function into a power series. Laplace uses complex variables when calculating difficult integrals and develops a method for solving linear, difference and differential equations known as the Laplace transform.

Beginning in 1799, works appeared in which more or less convenient interpretations of complex numbers were given and actions on them were defined. A fairly general theoretical interpretation and geometric interpretation was published by K. Gauss only in 1831.

L. Euler and his contemporaries left a rich legacy to their descendants in the form of accumulated, sometimes systematized, sometimes not, but still scattered facts on TFKP. We can say that the factual material on the functions of imaginary quantities seemed to require its systematization in the form of a theory. This theory began its development.

The third stage of the formation of TFKP (XIX centuries - XX centuries). The main achievements here belong to O. Cauchy (1789-1857), B. Riemann (1826-1866), and K. Weierstrass (1815-1897). Each of them represented one of the directions of development of TFKP.

The representative of the first direction, which in the history of mathematics was called “the theory of monogenic or differentiable functions,” was O. Cauchy. He formalized scattered facts on differential and integral calculus of functions of a complex variable, explained the meaning of basic concepts and operations with imaginary ones. In the works of O. Cauchy, the theory of limits and the theory of series and elementary functions based on it are presented, and a theorem is formulated that completely clarifies the region of convergence of a power series. In 1826, O. Cauchy introduced the term: deduction (literally: remainder). In his works from 1826 to 1829, he created the theory of residues. O. Cauchy derived an integral formula; obtained a theorem for the existence of the expansion of a function of a complex variable into power series (1831). O. Cauchy laid the foundations of the theory of analytic functions of many variables; determined the main branches of multivalued functions of a complex variable; first used plane cuts (1831-1847). In 1850, he introduced the concept of monodromic functions and identified the class of monogenic functions.

A follower of O. Cauchy was B. Riemann, who created his own “geometric” (second) direction of development of TFKP. In his works, he overcame the isolation of ideas about the functions of complex variables and formed new sections of this theory, closely related to other disciplines. Riemann made a significantly new step in the history of the theory of analytic functions; he proposed to associate with each function of a complex variable the idea of ​​mapping one domain onto another. He established the differences between the functions of a complex and a real variable. B. Riemann laid the foundation for the geometric theory of functions, introduced the Riemann surface, developed the theory of conformal mappings, established the connection between analytic and harmonic functions, and introduced the zeta function.

Further development of TFKP occurred in a different (third) direction. The basis of which was the possibility of representing functions by power series. This direction has been given the name “analytical” in history. It was formed in the works of K. Weierstrass, in which he brought to the fore the concept of uniform convergence. K. Weierstrass formulated and proved a theorem on the legality of bringing similar terms in a series. K. Weierstrass obtained a fundamental result: the limit of a sequence of analytic functions that uniformly converges inside a certain domain is an analytic function. He was able to generalize Cauchy's theorem on the power series expansion of a function of a complex variable and described the process of analytical continuation of power series and its application to the representation of solutions to a system of differential equations. K. Weierstrass established the fact of not only absolute convergence of the series, but also uniform convergence. Weierstrass's theorem on the expansion of an entire function into a product appears. He lays the foundations of the theory of analytic functions of many variables and builds a theory of divisibility of power series.

Let us consider the development of the theory of analytic functions in Russia. Russian mathematicians of the 19th century. for a long time they did not want to devote themselves to a new field of mathematics. Despite this, we can name several names to whom it was not alien, and list some of the works and achievements of these Russian mathematicians.

One of the Russian mathematicians was M.V. Ostrogradsky (1801-1861). About the research of M.V. Little is known about Ostrogradsky in the field of the theory of analytic functions, but O. Cauchy spoke with praise of this young Russian scientist who applied integrals and gave new proofs of formulas and generalized other formulas. M.V. Ostrogradsky wrote the work “Remarks on Definite Integrals”, in which he derived Cauchy’s formula for the subtraction of a function with respect to a pole of the nth order. He outlined applications of residue theory and Cauchy's formula to the evaluation of definite integrals in an extensive public lecture course given in 1858–1859.

A number of works by N.I. date back to the 1930s. Lobachevsky, which are of direct importance for the theory of functions of a complex variable. The theory of elementary functions of a complex variable is contained in his work “Algebra or the calculation of finite ones” (Kazan, 1834). In which cos x and sin x are initially determined for real x as real and

imaginary part of the function ex^. Using previously established properties of the exponential function and power expansions, all the basic properties of trigonometric functions are derived. By-

Apparently, Lobachevsky attached special importance to such a purely analytical construction of trigonometry, independent of Euclidean geometry.

It can be argued that in the last decades of the 19th century. and the first decade of the 20th century. Fundamental research on the theory of functions of a complex variable (F. Klein, A. Poincaré, P. Koebe) consisted of gradually clarifying that Lobachevsky geometry is at the same time the geometry of analytic functions of one complex variable.

In 1850, professor at St. Petersburg University (later academician) I.I. Somov (1815-1876) published “Foundations of the Theory of Analytical Functions,” which were based on Jacobi’s “New Foundations.”

However, the first truly “original” Russian researcher in the field of the theory of analytic functions of a complex variable was Yu.V. Sokhotsky (1842-1929). He defended his master's thesis “The Theory of Integral Residues with Some Applications” (St. Petersburg, 1868). Since the fall of 1868, Yu.V. Sokhotsky taught courses on the theory of functions of an imaginary variable and on continued fractions with applications to analysis. Master's thesis Yu.V. Sokhotsky is devoted to applications of the theory of residues to the inversion of power series (Lagrange series) and in particular to the expansion of analytic functions into continued fractions, as well as to Legendre polynomials. In this work, the famous theorem on the behavior of an analytic function in a neighborhood of an essentially singular point was formulated and proven. In Sokhotsky's doctoral dissertation

(1873) the concept of an integral of Cauchy type is introduced for the first time in expanded form: *r/ ^ & _ where

a and b are two arbitrary complex numbers. The integral is assumed to be taken along a certain curve (“trajectory”) connecting a and b. In this work, a number of theorems are proved.

The works of N.E. played a huge role in the history of analytical functions. Zhukovsky and S.A. Chaplygin, who opened up a vast area of ​​its applications in aero- and hydromechanics.

Speaking about the development of the theory of analytic functions, one cannot fail to mention the research of S.V. Kovalevskaya, although their main significance lies beyond the scope of this theory. The success of her work was due to a completely new formulation of the problem in terms of the theory of analytic functions and consideration of time t as a complex variable.

At the turn of the 20th century. The nature of scientific research in the field of the theory of functions of a complex variable is changing. If previously most of the research in this area was carried out in terms of the development of one of three directions (the theory of monogenic or differentiable Cauchy functions, geometric and physical ideas of Riemann, the analytical direction of Weierstrass), now the differences and associated disputes are overcome, appearing and growing rapidly the number of works in which a synthesis of ideas and methods is carried out. One of the main concepts on which the connection and correspondence of geometric concepts and the apparatus of power series was clearly revealed was the concept of analytic continuation.

At the end of the 19th century. The theory of functions of a complex variable includes an extensive set of disciplines: geometric theory of functions, based on the theory of conformal mappings and Riemann surfaces. We obtained a complete form of the theory of various types of functions: integer and meromorphic, elliptic and modular, automorphic, harmonic, algebraic. In close connection with the last class of functions, the theory of Abelian integrals was developed. Adjacent to this complex was the analytical theory of differential equations and the analytical theory of numbers. The theory of analytic functions established and strengthened connections with other mathematical disciplines.

The richness of the relationships between TFKP and algebra, geometry and other sciences, the creation of the systematic foundations of the science of TFKT itself, and its great practical significance contributed to the formation of TFKT as an educational subject. However, simultaneously with the completion of the formation of the foundations, new ideas were introduced into the theory of analytical functions that significantly changed its composition, nature and goals. Monographs appear containing a systematic presentation of the theory of analytic functions in a style close to axiomatic and also having educational purposes. Apparently, the significance of the results on TFKP obtained by scientists of the period under review encouraged them to popularize TFKP in the form of lecturing and publishing monographic studies from an educational perspective. It can be concluded that TFKP emerged as an educational

subject. In 1856, C. Briot and T. Bouquet published a small memoir, “Study of Functions of an Imaginary Variable,” which was essentially the first textbook. General concepts in the theory of functions of a complex variable began to be developed in lectures. Since 1856, K. Weierst-Rass lectured on the representation of functions by convergent power series, and since 1861 - on the general theory of functions. In 1876, a special essay by K. Weierstrass appeared: “On the Theory of Single-valued Analytical Functions,” and in 1880, “On the Doctrine of Functions,” in which his theory of analytic functions acquired a certain completeness.

Weierstrass's lectures served for many years as a prototype for textbooks on the theory of functions of a complex variable, which have since begun to appear quite often. It was in his lectures that basically the modern standard of rigor in mathematical analysis was built and the structure that became traditional was highlighted.

BIBLIOGRAPHICAL LIST

1. Andronov I.K. Mathematics of real and complex numbers. M.: Education, 1975.

2. Klein F. Lectures on the development of mathematics in the 19th century. M.: ONTI, 1937. Part 1.

3. Lavrentiev M.A., Shabat B.V. Methods of the theory of functions of a complex variable. M.: Nauka, 1987.

4. Markushevich A.I. Theory of analytic functions. M.: State. Publishing house of technical and theoretical literature, 1950.

5. Mathematics of the 19th century. Geometry. Theory of analytic functions / ed. A. N. Kolmogorov and A. P. Yushkevich. M.: Nauka, 1981.

6. Mathematical Encyclopedia / Ch. ed. I. M. Vinogradov. M.: Soviet Encyclopedia, 1977. T. 1.

7. Mathematical Encyclopedia / Ch. ed. I. M. Vinogradov. M.: Soviet Encyclopedia, 1979. T. 2.

8. Young V.N. Fundamentals of the doctrine of number in the 18th and early 19th centuries. M.: Uchpedgiz, 1963.

9. Rybnikov K.A. History of mathematics. M.: Moscow State University Publishing House, 1963. Part 2.

NOT. Lyakhova TOUCHING FLAT CURVES

The question of tangency of plane curves, in the case when the abscissas of common points are found from an equation of the form Pn x = 0, where P x ​​is some polynomial, is directly related to the question

on the multiplicity of roots of the polynomial Pn x. This article formulates the corresponding statements for the cases of explicit and implicit specification of functions whose graphs are curves, and also shows the application of these statements in solving problems.

If the curves that are graphs of the functions y = f(x) and y = ср x have a common point

M() x0; v0, i.e. y0 = f x0 =ср x0 and tangents to the indicated curves drawn at the point M() x0; v0 do not coincide, then they say that the curves y = fix) and y - ср x intersect at the point Mo xo;Uo

Figure 1 shows an example of the intersection of function graphs.

One of the ways to solve differential equations (systems of equations) with constant coefficients is the method of integral transformations, which allows the function of a real variable (the original function) to be replaced by a function of a complex variable (the image of the function). As a result, the operations of differentiation and integration in the space of original functions are transformed into algebraic multiplication and division in the space of image functions. One of the representatives of the integral transformation method is the Laplace Transform.

Continuous Laplace transform– an integral transformation that connects a function of a complex variable (function image) with a function of a real variable (original function). In this case, the function of a real variable must satisfy the following conditions:

The function is defined and differentiable on the entire positive half-axis of the real variable (the function satisfies the Dirichlet conditions);

The value of the function before the initial moment is equal to zero ;

The increase of the function is limited by the exponential function, i.e. for a function of a real variable there are such positive numbers M f(x) | With , What at , where c – abscissa of absolute convergence (some positive number).

Laplace transform (direct integral transform) a function of a real variable is called a function of the following form (function of a complex variable):

A function is called the original of a function, and a function is called its image. Complex variable is called the Laplace operator, where is the angular frequency and is some positive constant number.

As a first example, let's define an image for a constant function

As a second example, let's define an image for the cosine function . Taking into account Euler's formula, the cosine function can be represented as the sum of two exponentials .

In practice, to perform the direct Laplace transform, transformation tables are used, which present the originals and images of standard functions. Below are some of these features.

Original and image for exponential function

Original and image for cosine function

Original and image for sine function

Original and image for exponentially decaying cosine

Original and image for exponentially decaying sine

It should be noted that the function is a Heaviside function, which takes the value zero for negative values ​​of the argument and takes the value equal to one for positive values ​​of the argument.

Properties of the Laplace Transform

Linearity theorem

The Laplace transform has the property of linearity, i.e. any linear relationship between function originals is valid for images of these functions.

The linearity property simplifies finding the originals of complex images, since it allows the image of a function to be represented as a sum of simple terms, and then to find the originals of each represented term.

Original differentiation theorem functions

Differentiation of the original function corresponds to multiplication

For non-zero initial conditions:

At zero initial conditions (special case):

Thus, the operation of differentiating a function is replaced by an arithmetic operation in the image space of the function.

Original integration theorem functions

Integration of the original function corresponds division images of functions on the Laplace operator.

Thus, the operation of integrating a function is replaced by an arithmetic operation in the image space of the function.

Similarity theorem

Changing the argument of a function (compression or expansion of the signal) in the time domain leads to an inverse change in the argument and ordinate of the function image.

Increasing the pulse duration causes compression of its spectral function and a decrease in the amplitudes of the harmonic components of the spectrum.

Delay theorem

The delay (shift, displacement) of the signal according to the argument of the original function by an interval leads to a change in the phase-frequency function of the spectrum (phase angle of all harmonics) by a given value without changing the modulus (amplitude function) of the spectrum.

The resulting expression is valid for any

Displacement theorem

The delay (shift, displacement) of the signal by the argument of the function image leads to the multiplication of the original function by an exponential factor

From a practical point of view, the displacement theorem is used in defining images of exponential functions.

Convolution theorem

Convolution is a mathematical operation applied to two functions and , generating a third function. In other words, having the response of a certain linear system to an impulse, you can use convolution to calculate the system’s response to the entire signal.

Thus, the convolution of the originals of two functions can be represented as a product of the images of these functions. The verification theorem is used when considering transfer functions, when the system response (output signal from a four-port network) is determined when a signal is applied to the input of a four-port network with a pulse transient response.

Linear quadripole

Inverse Laplace transform

The Laplace transform is reversible, i.e. a function of a real variable is uniquely determined from a function of a complex variable . To do this, use the inverse Laplace transform formula(Mellin formula, Bromwich integral), which has the following form:

In this formula, the limits of integration mean that integration proceeds along an infinite straight line, which is parallel to the imaginary axis and intersects the real axis at point . Taking into account that the latter expression can be rewritten as follows:

In practice, to perform the inverse Laplace transform, the image of the function is decomposed into the sum of simple fractions by the method of undetermined coefficients and for each fraction (in accordance with the linearity property) the original function is determined, including taking into account the table of typical functions. This method is valid for depicting a function that is a proper rational fraction. It should be noted that the simplest fraction can be represented as a product of linear and quadratic factors with real coefficients depending on the type of roots of the denominator:

If there is a zero root in the denominator, the function is expanded into a fraction like:

If there is a zero n-fold root in the denominator, the function is expanded into a fraction like:

If there is a real root in the denominator, the function is expanded into a fraction like:

If there is a real n-fold root in the denominator, the function is expanded into a fraction like:

If there is an imaginary root in the denominator, the function is expanded into a fraction like:

In the case of complex conjugate roots in the denominator, the function is expanded into a fraction like:

∙ In general if the image of a function is a proper rational fraction (the degree of the numerator is less than the degree of the denominator of the rational fraction), then it can be decomposed into a sum of simple fractions.

∙ In a special case if the denominator of the image of a function is decomposed only into simple roots of the equation, then the image of the function can be expanded into a sum of simple fractions as follows:

Unknown coefficients can be determined by the unknown coefficients method or a simplified method using the following formula:

The value of the function at the point ;

The value of the derivative of the function at the point.

To solve linear differential equations we will use the Laplace transform.

Laplace transform called the ratio

putting functions x(t) real variable t matching function X(s) complex variable s (s = σ+ jω). Wherein x(t) called original, X(s)- image or Laplace image And s- Laplace transform variable. The original is denoted in lowercase, and its image is denoted by the same capital letter.

It is assumed that the function x(t), undergoing the Laplace transform, has the following properties:

1) function x(t) defined and piecewise differentiable on the interval )