Image from original Laplace transform. Laplace transform basic property definitions are the Duhamel formula. Direct Laplace transform
To solve linear differential equations we will use the Laplace transform.
Laplace transform call the ratio
assigning functions x (t) real variable t match function X (s) complex variable s (s = σ+ jω). Wherein x (t) are called original, X (s)- image or Laplace image and s- Laplace transform variable. The original is denoted in lowercase, and its image is in the capitalized letter of the same name.
It is assumed that the function x(t) subject to the Laplace transform has the following properties:
1) function x (t) is defined and piecewise differentiable on the interval where< b. Разобьём отрезок [, b ] с помощью точек деления на n элементарных
8 Lecture 7 OPERATOR FUNCTIONS OF CIRCUITS Operator input and transfer functions Poles and zeros of circuit functions 3 Conclusions Operator input and transfer functions An operator function of a chain is a relation
68 Lecture 7 TRANSITION PROCESSES IN FIRST ORDER CIRCUITS Plan 1 Transient processes in RC-circuits of the first order 2 Transient processes in R-circuits of the first order 3 Examples of calculation of transient processes in circuits
4 LINEAR ELECTRIC CIRCUITS OF AC SINUSOIDAL CURRENT AND METHODS OF THEIR CALCULATION 4.1 ELECTRIC MACHINES. SINUSOIDAL CURRENT GENERATION PRINCIPLE 4.1.012. Sinusoidal current is called instantaneous
Federal Agency for Education State Educational Institution of Higher Professional Education "KUBAN STATE UNIVERSITY" Faculty of Physics and Technology Department of Optoelectronics
~ ~ FKP Derivative of the function of a complex variable FKP of the Cauchy - Riemann condition the concept of regularity of the FKP Image and form of a complex number Form of the FKP: where the real function of two variables is real
Section II. Mathematical analysis
E. Yu. Anokhina
THE HISTORY OF DEVELOPMENT AND THE FORMATION OF THE THEORY OF THE FUNCTION OF A COMPLEX VARIABLE (TFKP) BY A EDUCATIONAL SUBJECT
One of the difficult mathematics courses is the TFKP course. The complexity of this course is due, first of all, to the variety of its interrelationships with other mathematical disciplines, historically expressed in the broad applied direction of the TFKP science.
In the scientific literature on the history of mathematics, there are scattered information about the history of the development of TFKP, they require systematization and generalization.
In this regard, the main task of this article is to briefly describe the development of TFKP and the formation of this theory as an academic subject.
As a result of the study, the following three stages in the development of TFKP as a science and academic subject were identified:
The stage of emergence and recognition of complex numbers;
The stage of accumulation of factual material according to the functions of imaginary values;
Formation stage of the theory of functions of a complex variable.
The first stage in the development of the 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 about algebraic rules). The work of J. Cardano had the main task of substantiating general algebraic methods for solving equations of the third and fourth degrees, not long before that discovered by Ferro (1465-1526), Tartaglia (1506-1559) and Ferrari (1522-1565). If the cubic equation is reduced to the form
x3 + px + q = 0,
and there should be
When (μ ^ Ap V (| - 70 the equation has three real roots, two of them
are equal to each other. If then the equation has one real and two co-
twisted complex root. Complex numbers appear in the final result, so J. Cardano could have done what he did before him: declare the equation to have
one root. When (<7 Г + (р V < (). тогда уравнение имеет три действительных корня. Этот так
the called irreducible case is characterized by one peculiarity that was not encountered until the 16th century. The equation x3 - 21x + 20 = 0 has three real roots 1, 4, - 5 which is easy
make sure 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 is not the result here, but an intermediate term in the calculations that lead to the real roots of the equation in question. J. Cardano faced a difficulty and realized that in order to preserve the generality of this formula, it is necessary to abandon the complete ignorance of complex numbers. J. D'Alembert (1717-1783) believed that it was precisely this circumstance that made J. Cardano and the mathematicians who followed this idea become 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 unlimited use of complex numbers by anything. The second was 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 was ignored by Descartes. 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 a 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 topical. This was the beginning of a new stage in the development of TFKP.
The second stage in the development of the TFKP (early 18th century - 19th century). In the XVIII century. L. Euler expressed the idea of the algebraic closedness of the field of complex numbers. 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 the proper 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 infinitesimal" introduced a complex variable as the most general concept of a variable, using complex numbers in the expansion of functions into linear factors. L. Euler is rightfully considered one of the creators of the TFKP. In the works of L. Euler, elementary functions of a complex variable (1740-1749) were studied in detail, conditions for differentiability (1755) and the beginning of the integral calculus of functions of a complex variable (1777) were given. L. Euler practically introduced conformal mapping (1777). He called these mappings "similar in small", and the term "conformal" was first used, apparently, by the St. Petersburg academician F. Schubert (1789). L. Euler also led numerous applications of functions of a complex variable to various mathematical problems and laid the foundation for their application in hydrodynamics (17551757) and cartography (1777). K. Gauss formulates the definition of an integral in the complex plane, an integral theorem on the expansion of an analytic function in a power series. Laplace uses complex variables to calculate difficult integrals and develops a method for solving linear, difference and differential equations known as the Laplace transform.
Since 1799, works have appeared in which more or less convenient interpretations of a complex number are given and actions on them are 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 descendants in the form of accumulated, somewhere systematized, somewhere not, but still scattered facts on the TFKP. We can say that the factual material on the functions of imaginary quantities, as it were, required its systematization in the form of a theory. This theory began its formation.
The third stage of the formation of the TFKP (XIX century - XX century). 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 the TFKP development.
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 the differential and integral calculus of functions of a complex variable, clarified the meaning of the 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 stated, a theorem is formulated that completely clarifies the domain of convergence of a power series. In 1826, O. Cauchy introduced the term: deduction (literally: remainder). In his writings from 1826 to 1829, he created the theory of deductions. O. Cauchy derived an integral formula; obtained an existence theorem for an expansion of a function of a complex variable in power series (1831). O. Cauchy laid the foundations for the theory of analytic functions of several 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, distinguished the class of monogenic functions.
A follower of O. Cauchy was B. Riemann, who also created his own "geometric" (second) direction of the TFKP development. In his works, he overcame the isolation of ideas about functions of complex variables and formed new departments of this theory, closely related to other disciplines. Riemann made a significant new step in the history of the theory of analytic functions, he proposed to connect the representation of the mapping of one region to another with each function of a complex variable. He established a distinction 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 a connection between analytic and harmonic functions, and introduced the zeta function into consideration.
Further development of TFKP took place in a different (third) direction. The basis of which was the possibility of representing functions by power series. The name “analytical” has been stuck in history for this direction. 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 the reduction of similar terms in a series. K. Weierstrass obtained a fundamental result: the limit of a sequence of analytic functions converging uniformly inside a certain domain is an analytic function. He was able to generalize the Cauchy theorem on the expansion in a power series of a function of a complex variable and described the process of analytic continuation of power series and its application to the representation of solutions of a system of differential equations. K. Weierstrass established the fact not only of the absolute convergence of the series, but also of the uniform convergence. The Weierstrass theorem on the expansion of an entire function into a product appears. He lays the foundations for the theory of analytic functions of several variables, constructs the theory of divisibility of power series.
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, one can name several names for which she 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). On the research of M.V. Little is known about Ostrogradskii 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. Ostrogradskiy wrote a paper "Remarks on definite integrals", in which he derived the Cauchy formula for the residue of a function with respect to the n-th order pole. He outlined the application of the theory of residues and the Cauchy formula to the calculation of definite integrals in an extensive public lecture course given in 1858-1859.
A number of works by N.I. 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 calculation of finite" (Kazan, 1834). In which cos x and sin x are determined initially for x real as real and
the imaginary part of the function ex ^. Using the previously established properties of the exponential function and power expansions, all the main properties of trigonometric functions are derived. By-
Apparently, Lobachevsky attached particular importance to such a purely analytical construction of trigonometry, independent of Euclidean geometry.
It can be argued that in the last decades of the XIX century. and the first decade of the XX century. fundamental research in the theory of functions of a complex variable (F. Klein, A. Poincaré, P. Kebe) consisted in the gradual clarification of the fact that the geometry of Lobachevsky is at the same time the geometry of analytic functions of one complex variable.
In 1850, Professor of St. Petersburg University (later Academician) I.I. Somov (1815-1876) published The Foundations of the Theory of Analytic 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 gave courses in the theory of functions of an imaginary variable and on continued fractions with applications to analysis. Master's thesis Yu.V. Sokhotskii is devoted to applications of the theory of residues to the inversion of a power series (Lagrange series) and, in particular, to the expansion of analytic functions in continued fractions, as well as to Legendre polynomials. In this paper, the famous theorem on the behavior of an analytic function in a neighborhood of an essentially singular point is formulated and proved. Sokhotskiy's doctoral dissertation
(1873) introduced for the first time in expanded form the concept of a Cauchy-type integral: * y ^ & _ where
a and b are two arbitrary complex numbers. The integral is assumed to be taken along some curve ("trajectory") connecting a and b. In this paper, a number of theorems are proved.
A huge role in the history of analytical functions was played by the works of N.E. Zhukovsky and S.A. Chaplygin, who discovered the boundless field of its applications in aerodynamics and hydromechanics.
Speaking about the development of the theory of analytic functions, one cannot but mention the studies of S.V. Kovalevskaya, although their main significance lies outside the boundaries 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 the consideration of the time t as a complex variable.
At the turn of the XX century. the nature of scientific research in the field of the theory of functions of a complex variable is changing. If earlier 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, the geometric and physical ideas of Riemann, the analytical direction of Weierstrass), now the differences and related disputes are being overcome, there is and is growing rapidly the number of works in which the synthesis of ideas and methods is carried out. One of the basic concepts on which the connection and correspondence of geometric representations and the apparatus of power series was clearly revealed was the concept of analytic continuation.
At the end of the XIX century. the theory of functions of a complex variable includes a wide range of disciplines: geometric theory of functions based on the theory of conformal mappings and Riemann surfaces. Received an integral form of the theory of various types of functions: entire and meromorphic, elliptic and modular, automorphic, harmonic, algebraic. In close connection with the latter class of functions, the theory of Abelian integrals was developed. Analytical theory of differential equations and analytical theory of numbers were adjacent to this complex. The theory of analytic functions has established and strengthened links with other mathematical disciplines.
The richness of the interrelationships of the TFKP with algebra, geometry and other sciences, the creation of the systematic foundations of the science of the TFKP itself, its great practical significance contributed to the formation of the TFKP as an academic subject. However, simultaneously with the completion of the formation of the foundations, new ideas were introduced into the theory of analytical functions, significantly changing its composition, nature and goals. Monographs appear that contain a systematic presentation of the theory of analytic functions in a style close to the axiomatic and also have educational goals. Apparently, the significance of the results on TFKP, obtained by scientists of the period under review, prompted them to popularize TFKP in the form of lecturing and publishing monographic studies in a teaching perspective. We can conclude about the emergence of TFKP as an educational
subject. In 1856 C. Briot and T. Bouquet published a short memoir "Investigation of the functions of an imaginary variable", which is essentially the first textbook. General concepts in the theory of functions of a complex variable began to be developed in lectures. Since 1856 K. Weierht-rass has lectured on the representation of functions by convergent power series, and since 1861 - on the general theory of functions. In 1876, a special work by K. Weierstrass appeared: "On the theory of single-valued analytic functions", and in 1880 "On the doctrine of functions", in which his theory of analytic functions acquired a certain completeness.
Weierstrass' lectures served for many years as the prototype of textbooks on the theory of functions of a complex variable, which began to appear quite often since then. It was in his lectures that basically the modern standard of rigor in mathematical analysis was built and the structure that has become traditional was highlighted.
BIBLIOGRAPHIC LIST
1. Andronov I.K. Mathematics of real and complex numbers. Moscow: Education, 1975.
2. Klein F. Lectures on the development of mathematics in the 19th century. M .: ONTI, 1937. Part 1.
3. Lavrent'ev M.A., Shabat B.V. Methods of the theory of functions of a complex variable. Moscow: Nauka, 1987.
4. Markushevich A.I. Theory of analytic functions. M .: State. publishing house of technical and theoretical literature, 1950.
5. Mathematics of the XIX century. Geometry. Theory of analytic functions / ed. A.N. Kolmogorov and A.P. Yushkevich. Moscow: Nauka, 1981.
6. Encyclopedia of Mathematics / Ch. ed. I. M. Vinogradov. Moscow: Soviet Encyclopedia, 1977.Vol. 1.
7. Mathematical encyclopedia / Ch. ed. I. M. Vinogradov. Moscow: Soviet Encyclopedia, 1979.Vol. 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. Moscow: Moscow State University Publishing House, 1963. Part 2.
NOT. Lyakhova TOUCHING PLANE 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 the roots of the polynomial Pn x. In this article, the corresponding statements are formulated for the cases of explicit and implicit assignment of functions whose graphs are curves, and the application of these statements in solving problems is shown.
If the curves that are graphs of the functions y = f (x) and y = cp x have a common point
M () x0; v0, i.e. y0 = f x0 = cp 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 - cp x intersect at the point Mo xo; Yo
Figure 1 shows an example of intersection of graphs of functions.
This is the name of another type of integral transforms, 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.
Complex frequency concept.
Spectral methods, as is already known, are based on the fact that the signal under investigation is represented as a sum of an infinitely large number of elementary terms, each of which periodically changes in time according to the law.
The natural generalization of this principle lies in the fact that instead of complex exponential signals with purely imaginary indicators, exponential signals of the form are introduced into consideration, where is a complex number: called the complex frequency.
Two such complex signals can be used to compose a real signal, for example, according to the following rule:
where is the complex conjugate value.
Indeed, in this case
Depending on the choice of the real and imaginary parts of the complex frequency, various real signals can be obtained. So, if, but you get the usual harmonic oscillations of the form If, then, depending on the sign, you 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 a complex frequency turns out to be very useful, first of all, because it makes it possible, without resorting to generalized functions, to obtain spectral representations of signals whose mathematical models are not integrable.
Rice. 2.10. Real signals corresponding to different values of the complex frequency
Another consideration is also essential: exponential signals of the form (2.53) serve as a "natural" means of studying oscillations in various linear systems. These questions will be explored in Ch. eight.
It should be noted that the true physical frequency is 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 time values. The Laplace transform of this signal is a function of a complex variable given by an integral:
The signal is called the original, and the function is called its Laplace image (for short, just the image).
The condition that ensures the existence of the integral (2.54) is as follows: the signal must have no more than an exponential growth rate, i.e., 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 the Number a is called the abscissa of absolute convergence.
The variable in the main 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 zero at Thus, the Laplace transform can be considered
Just as it is done in the theory of Fourier transform, it is possible, knowing the image, to restore the original. For this, in the inverse Fourier transform formula
an analytical continuation should be performed, passing from the imaginary variable to the complex argument a. On the plane of the complex frequency, the integration is carried out along an infinitely long vertical axis located to the right of the abscissa of absolute convergence. Since at is the differential, the formula for the inverse Laplace transform takes the form
In the theory of functions of a complex variable, it is proved that Laplace images have "good" properties from the point of view 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), one can use flexible methods of the theory of residues.
In practice, Laplace transform tables are widely used, which collect information about the correspondence between the originals. and images. The presence of tables made the Laplace transform method popular both in theoretical studies and in engineering calculations of radio engineering devices and systems. In the Appendices to there is such a table that allows you to solve a fairly wide range of problems.
Examples of calculating Laplace transforms.
Image computation methods have much in common with what has already been studied in relation to the Fourier transform. Let's consider the most typical cases.
Example 2.4, Image of the 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 will vanish when the upper limit is substituted. As a result, we get the correspondence
As a special case of formula (2.56), you can find the image of a real exponential video pulse:
and a complex exponential signal:
Finally, putting in (2.57), we find the image of the Heaviside function:
Example 2.5. Delta function image.
Laplace transform- integral transformation linking the function F (s) (\ displaystyle \ F (s)) complex variable ( image) with the function f (x) (\ displaystyle \ f (x)) real variable ( original). With its help, the properties of dynamical systems are investigated and differential and integral equations are solved.
One of the features of the Laplace transform, which predetermined its widespread use in scientific and engineering calculations, is that many ratios and operations on originals correspond to simpler ratios over their images. Thus, the convolution of two functions is reduced in the image space to the operation of multiplication, and linear differential equations become algebraic.
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Subtitles
Definition
Direct Laplace transform
lim b → ∞ ∫ 0 b | f (x) | e - σ 0 x d x = ∫ 0 ∞ | f (x) | e - σ 0 xdx, (\ 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 analytic function for σ ⩾ 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)). Precise bottom edge σ a (\ displaystyle \ sigma _ (a)) sets of numbers σ (\ displaystyle \ sigma), under which this condition is satisfied, is called abscissa of absolute convergence Laplace transform for the function.
- 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 the following cases:
- σ ⩾ 0 (\ displaystyle \ sigma \ geqslant 0): the Laplace transform exists if there is an integral ∫ 0 ∞ | f (x) | d x (\ displaystyle \ int \ limits _ (0) ^ (\ infty) | f (x) | \, dx);
- σ> σ a (\ displaystyle \ sigma> \ sigma _ (a)): the 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 end x 1> 0 (\ displaystyle x_ (1)> 0) and | 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) or σ> σ a (\ displaystyle \ sigma> \ sigma _ (a))(which of the bounds is greater): the Laplace transform exists if there is a Laplace transform for the function f ′ (x) (\ displaystyle f "(x))(derived from f (x) (\ displaystyle f (x))) for σ> σ a (\ displaystyle \ sigma> \ sigma _ (a)).
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:
- If the image F (s) (\ displaystyle F (s))- analytical function for σ ⩾ σ a (\ displaystyle \ sigma \ geqslant \ sigma _ (a)) and has 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) for 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))) analytic about each 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 F k (s) = L (fk (x)) (σ> σ ak: k = 1, 2,…, n) (\ displaystyle F_ (k) (s) = (\ mathcal (L)) \ (f_ (k) (x) \) \; \; (\ sigma> \ sigma _ (ak) \ colon k = 1, \; 2, \; \ ldots, \; n)), then the inverse transformation exists and the corresponding forward transformation has the absolute convergence abscissa.
Note: these are sufficient conditions for existence.
- Convolution theorem
Main article: Convolution theorem
- Differentiating and integrating the original
The Laplace image of the first derivative of the original with respect to the argument is the product of the image by the argument of the latter minus the original at zero on the right:
L (f ′ (x)) = s ⋅ F (s) - f (0 +). (\ displaystyle (\ mathcal (L)) \ (f "(x) \) = s \ cdot F (s) -f (0 ^ (+)).)Initial and final value theorems (limit theorems):
f (∞) = lim s → 0 s F (s) (\ displaystyle f (\ infty) = \ lim _ (s \ to 0) sF (s)) if all poles of the function s F (s) (\ displaystyle sF (s)) 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 dynamical system.
- Other properties
Linearity:
L (a f (x) + b g (x)) = a F (s) + b G (s). (\ displaystyle (\ mathcal (L)) \ (af (x) + bg (x) \) = aF (s) + bG (s).)Multiplication by a number:
L (f (a x)) = 1 a F (s a). (\ displaystyle (\ mathcal (L)) \ (f (ax) \) = (\ frac (1) (a)) F \ left ((\ frac (s) (a)) \ right).)Direct and inverse Laplace transform of some functions
Below is a table of Laplace transform for some functions.
| № | Function | Time domain x (t) = L - 1 (X (s)) (\ displaystyle x (t) = (\ mathcal (L)) ^ (- 1) \ (X (s) \)) |
Frequency domain X (s) = L (x (t)) (\ displaystyle X (s) = (\ mathcal (L)) \ (x (t) \)) |
Convergence region for causal systems |
|---|---|---|---|---|
| 1 | perfect lag | δ (t - τ) (\ displaystyle \ delta (t- \ tau) \) | e - τ s (\ displaystyle e ^ (- \ tau s) \) | |
| 1a | single impulse | δ (t) (\ displaystyle \ delta (t) \) | 1 (\ displaystyle 1 \) | ∀ s (\ displaystyle \ forall s \) |
| 2 | lag n (\ displaystyle n) | (t - τ) n n! e - α (t - τ) ⋅ H (t - τ) (\ displaystyle (\ frac ((t- \ tau) ^ (n)) (n}e^{-\alpha (t-\tau)}\cdot H(t-\tau)} !} | e - τ s (s + α) n + 1 (\ displaystyle (\ frac (e ^ (- \ tau s)) ((s + \ alpha) ^ (n + 1)))) | s> 0 (\ displaystyle s> 0) |
| 2a | sedate n (\ displaystyle n)-th order | t n n! ⋅ H (t) (\ displaystyle (\ frac (t ^ (n)) (n}\cdot H(t)} !} | 1 s n + 1 (\ displaystyle (\ frac (1) (s ^ (n + 1)))) | s> 0 (\ displaystyle s> 0) |
| 2a.1 | sedate q (\ displaystyle q)-th order | t q Γ (q + 1) ⋅ H (t) (\ displaystyle (\ frac (t ^ (q)) (\ Gamma (q + 1))) \ cdot H (t)) | 1 s q + 1 (\ displaystyle (\ frac (1) (s ^ (q + 1)))) | s> 0 (\ displaystyle s> 0) |
| 2a.2 | unit function | H (t) (\ displaystyle H (t) \) | 1 s (\ displaystyle (\ frac (1) (s))) | s> 0 (\ displaystyle s> 0) |
| 2b | lag unit function | H (t - τ) (\ displaystyle H (t- \ tau) \) | e - τ s s (\ displaystyle (\ frac (e ^ (- \ tau s)) (s))) | s> 0 (\ displaystyle s> 0) |
| 2c | Speed step | t ⋅ H (t) (\ displaystyle t \ cdot H (t) \) | 1 s 2 (\ displaystyle (\ frac (1) (s ^ (2)))) | s> 0 (\ displaystyle s> 0) |
| 2d | n (\ displaystyle n)-th order with frequency shift | t n n! e - α t ⋅ H (t) (\ displaystyle (\ frac (t ^ (n)) (n}e^{-\alpha t}\cdot H(t)} !} | 1 (s + α) n + 1 (\ displaystyle (\ frac (1) ((s + \ alpha) ^ (n + 1)))) | s> - α (\ displaystyle s> - \ alpha) |
| 2d.1 | exponential decay | e - α t ⋅ H (t) (\ displaystyle e ^ (- \ alpha t) \ cdot H (t) \) | 1 s + α (\ displaystyle (\ frac (1) (s + \ alpha))) | s> - α (\ displaystyle s> - \ alpha \) |
| 3 | exponential approximation | (1 - e - α t) ⋅ H (t) (\ displaystyle (1-e ^ (- \ alpha t)) \ cdot H (t) \) | α s (s + α) (\ displaystyle (\ frac (\ alpha) (s (s + \ alpha)))) | s> 0 (\ displaystyle s> 0 \) |
| 4 | sinus | sin (ω t) ⋅ H (t) (\ displaystyle \ sin (\ omega t) \ cdot H (t) \) | ω s 2 + ω 2 (\ displaystyle (\ frac (\ omega) (s ^ (2) + \ omega ^ (2)))) | s> 0 (\ displaystyle s> 0 \) |
| 5 | cosine | cos (ω t) ⋅ H (t) (\ displaystyle \ cos (\ omega t) \ cdot H (t) \) | s s 2 + ω 2 (\ displaystyle (\ frac (s) (s ^ (2) + \ omega ^ (2)))) | s> 0 (\ displaystyle s> 0 \) |
| 6 | hyperbolic sine | s h (α t) ⋅ H (t) (\ displaystyle \ mathrm (sh) \, (\ alpha t) \ cdot H (t) \) | α s 2 - α 2 (\ displaystyle (\ frac (\ alpha) (s ^ (2) - \ alpha ^ (2)))) | s> | α | (\ displaystyle s> | \ alpha | \) |
| 7 | hyperbolic cosine | c h (α t) ⋅ H (t) (\ displaystyle \ mathrm (ch) \, (\ alpha t) \ cdot H (t) \) | s s 2 - α 2 (\ displaystyle (\ frac (s) (s ^ (2) - \ alpha ^ (2)))) | s> | α | (\ displaystyle s> | \ alpha | \) |
| 8 | exponentially decaying sinus |
e - α t sin (ω t) ⋅ H (t) (\ displaystyle e ^ (- \ alpha t) \ sin (\ omega t) \ cdot H (t) \) | ω (s + α) 2 + ω 2 (\ displaystyle (\ frac (\ omega) ((s + \ alpha) ^ (2) + \ omega ^ (2)))) | s> - α (\ displaystyle s> - \ alpha \) |
| 9 | exponentially decaying cosine |
e - α t cos (ω t) ⋅ H (t) (\ displaystyle e ^ (- \ alpha t) \ cos (\ omega t) \ cdot H (t) \) | s + α (s + α) 2 + ω 2 (\ displaystyle (\ frac (s + \ alpha) ((s + \ alpha) ^ (2) + \ omega ^ (2)))) | s> - α (\ displaystyle s> - \ alpha \) |
| 10 | root n (\ displaystyle n)-th order | t n ⋅ H (t) (\ displaystyle (\ sqrt [(n)] (t)) \ cdot H (t)) | s - (n + 1) / n ⋅ Γ (1 + 1 n) (\ displaystyle s ^ (- (n + 1) / n) \ cdot \ Gamma \ left (1 + (\ frac (1) (n) ) \ right)) | s> 0 (\ displaystyle s> 0) |
| 11 | natural logarithm | ln (t t 0) ⋅ H (t) (\ displaystyle \ ln \ left ((\ frac (t) (t_ (0))) \ right) \ cdot H (t)) | - t 0 s [ln (t 0 s) + γ] (\ displaystyle - (\ frac (t_ (0)) (s)) [\ ln (t_ (0) s) + \ gamma]) | s> 0 (\ displaystyle s> 0) |
| 12 | Bessel function first kind order n (\ displaystyle n) |
J n (ω t) ⋅ H (t) (\ displaystyle J_ (n) (\ omega t) \ cdot H (t)) | ω n (s + s 2 + ω 2) - ns 2 + ω 2 (\ displaystyle (\ frac (\ omega ^ (n) \ left (s + (\ sqrt (s ^ (2) + \ omega ^ (2) )) \ right) ^ (- n)) (\ sqrt (s ^ (2) + \ omega ^ (2))))) | s> 0 (\ displaystyle s> 0 \) (n> - 1) (\ displaystyle (n> -1) \) |
| 13 | first kind order n (\ displaystyle n) |
I n (ω t) ⋅ H (t) (\ displaystyle I_ (n) (\ omega t) \ cdot H (t)) | ω n (s + s 2 - ω 2) - ns 2 - ω 2 (\ displaystyle (\ frac (\ omega ^ (n) \ left (s + (\ sqrt (s ^ (2) - \ omega ^ (2) )) \ right) ^ (- n)) (\ sqrt (s ^ (2) - \ omega ^ (2))))) | s> | ω | (\ displaystyle s> | \ omega | \) |
| 14 | Bessel function second kind zero order |
Y 0 (α t) ⋅ H (t) (\ displaystyle Y_ (0) (\ alpha t) \ cdot H (t) \) | - 2 arsh (s / α) π s 2 + α 2 (\ displaystyle - (\ frac (2 \ mathrm (arsh) (s / \ alpha)) (\ pi (\ sqrt (s ^ (2) + \ alpha ^ (2)))))) | s> 0 (\ displaystyle s> 0 \) |
| 15 | modified Bessel function second kind, zero order |
K 0 (α t) ⋅ H (t) (\ displaystyle K_ (0) (\ alpha t) \ cdot H (t)) | ||
| 16 | error function | e r f (t) ⋅ H (t) (\ displaystyle \ mathrm (erf) (t) \ cdot H (t)) | e s 2/4 e r f c (s / 2) s (\ displaystyle (\ frac (e ^ (s ^ (2) / 4) \ mathrm (erfc) (s / 2)) (s))) | s> 0 (\ displaystyle s> 0) |
Notes to the table:
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