Laplace transform with graphs of functions. 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.
Two such complex signals can be used to compose a real signal, for example, according to the following rule:
where is the complex conjugate quantity.
Indeed, while
Depending on the choice of the real and imaginary parts of the complex frequency, various real signals can be obtained. So, if , but ordinary harmonic oscillations of the form If then, depending on the sign, either increasing or decreasing exponential oscillations are obtained. Such signals acquire a more complex form when . Here, the multiplier describes an envelope that changes exponentially with 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 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 Chap. 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 o of the complex frequency.
Basic ratios.
Let - some signal, real or complex, defined for t > 0 and equal to zero for negative values of time. The Laplace transform of this signal is a function of the 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 the integral (2.54) is as follows: the signal must have at most an exponential growth rate for ie 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, for 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 the Fourier transform, it is possible, knowing the image, to restore the original. To do this, in the formula for the inverse Fourier transform
an analytic continuation should be performed by passing from the imaginary variable to the complex argument a On the plane of the complex frequency, integration is carried out along an unlimitedly extended vertical axis located to the right of the abscissa of absolute convergence. Since for 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 are usually 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 made the Laplace transform method popular both in theoretical studies and in engineering calculations of radio engineering devices and systems. In the Annexes to there is such a table, which allows solving a fairly wide range of problems.
Examples of calculating Laplace transforms.
There are many similarities in the methods of computing images with what has already been studied in relation to the Fourier transform. Let's consider the most typical cases.
Example 2.4, Image of a 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 vanishes when the upper limit is substituted. As a result, we get the correspondence
As a special case of formula (2.56), one can find the 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. An image of a delta function.
Laplace transform- integral transformation relating 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 studied 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 on their images. Thus, the convolution of two functions in the space of images is reduced 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 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 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)). Exact lower bound 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 the integral exists ∫ 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 finite 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): a Laplace transform exists if a Laplace transform exists for the function f ′ (x) (\displaystyle f"(x))(derivative of 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 that the following conditions are met:
- If the image F (s) (\displaystyle F(s))- analytical function for σ ≥ 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) 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))) is analytic with respect to each z k (\displaystyle z_(k)) and equals zero for z 1 = z 2 = … = z n = 0 (\displaystyle z_(1)=z_(2)=\ldots =z_(n)=0), 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)), then the inverse transformation exists and the corresponding direct conversion has an abscissa of absolute convergence.
Note: these are sufficient conditions for existence.
- Convolution theorem
Main article: Convolution theorem
- Differentiation and integration of the original
According to Laplace, the first derivative of the original with respect to the argument is the product of the image and 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 with 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).)Multiply by 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 the Laplace transform table 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 area for causal systems |
|---|---|---|---|---|
| 1 | ideal lag | δ (t − τ) (\displaystyle \delta (t-\tau)\ ) | e − τ s (\displaystyle e^(-\tau s)\ ) | |
| 1a | single pulse | 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 | power 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 | power 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 | single function | H (t) (\displaystyle H(t)\ ) | 1 s (\displaystyle (\frac (1)(s))) | s > 0 (\displaystyle s>0) |
| 2b | single function with delay | 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) − n s 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) − n s 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 a r s h (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) |
Table notes:
| ||||
Section II. Mathematical analysis
E. Yu. Anokhina
HISTORY OF DEVELOPMENT AND FORMATION OF THE THEORY OF THE FUNCTION OF A COMPLEX VARIABLE (TFV) AS A SUBJECT
One of the complex mathematical courses is the TFKT course. The complexity of this course is due, first of all, to the diversity of its interrelationships with other mathematical disciplines, historically expressed in the broad applied orientation of the science of TFKT.
In the scientific literature on the history of mathematics, there is scattered information about the history of the development of the TFCT, they require systematization and generalization.
For this reason, the main goal of this article is short description development of the 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 TFCT as a science and academic subject were identified:
The stage of emergence and recognition of complex numbers;
The stage of accumulation of factual material on the functions of imaginary quantities;
The stage of formation 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 Artis magnae sive de regulis algebraitis (Great Art, or on algebraic rules). The work of G. Cardano had the main task of substantiating the general algebraic methods 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 + q = 0,
and should be
When (p^Ap 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-
spun complex roots. Complex numbers appear in the final result, so G. Cardano could do as they did before him: 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
check with a 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. G. Cardano encountered a difficulty and realized that in order to preserve the generality of this formula, it is necessary to abandon the complete disregard for complex numbers. J. d'Alembert (1717-1783) believed that it was precisely this circumstance that made G. 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 issue of recognizing the need for unrestricted use of complex numbers. The second - Descartes, who denied the possibility of interpreting complex numbers. Opposite to the opinion of Descartes 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 on 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 topical. This was the beginning of a new stage in the development of the TFCT.
The second stage in the development of the TFKP (the beginning of the 18th century - the 19th century). In the XVIII century. L. Euler expressed the idea of the algebraic closure of the field of complex numbers. The algebraic closure of the field of complex numbers C led mathematicians to the following conclusions:
That the study of functions and mathematical analysis in general acquire their proper completeness and completeness only when considering the behavior of functions in the 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, using complex numbers when decomposing functions into linear factors. L. Euler is rightfully considered one of the creators of the TFCT. In the works of L. Euler, elementary functions of a complex variable were studied in detail (1740-1749), conditions for differentiability (1755) and the beginning of the integral calculus of functions of a complex variable (1777) were given. L. Euler practically introduced the conformal mapping (1777). He called these mappings "similar in a small way", and the term "conformal" was first used, apparently, by the St. Petersburg academician F. Schubert (1789). L. Euler also gave 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 into 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.
Starting from 1799, papers appear in which more or less convenient interpretations of the 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 heritage to posterity in the form of accumulated, somewhere systematized, somewhere not, but still scattered facts on the TFCT. 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 has begun to take shape.
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 development of the 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 disparate facts on the differential and integral calculus of functions of a complex variable, explained 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 elucidates the region of convergence of a power series. In 1826, O. Cauchy introduced the term: deduction (literally: remainder). In writings from 1826 to 1829, he created the theory of deductions. O. Cauchy deduced the integral formula; obtained an existence theorem for the expansion of a function of a complex variable into power series (1831). O. Cauchy laid the foundations for the theory of analytic functions of several variables; determined the main branches of multi-valued functions of a complex variable; first used plane cuts (1831-1847). In 1850 he introduces the concept of monodromic functions and singles out the class of monogenic functions.
O. Cauchy's follower was B. Riemann, who also created his own "geometric" (second) direction of development of the TFCT. 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 an essentially 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 a mapping from one region to another. He distinguished 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, introduced the zeta function into consideration.
Further development of TFKP took place in another (third) direction. The basis of which was the possibility of representing functions by power series. This trend 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 reducing similar terms in a series. K. Weierstrass obtained a fundamental result: the limit of a sequence of analytic functions that converges uniformly inside a certain domain is an analytic function. He managed to generalize Cauchy's theorem on power series expansion 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 to a system of differential equations. K. Weierstrass established the fact of not only the absolute convergence of the series, but also the uniform convergence. The Weierstrass theorem appears on the expansion of an entire function into a product. He lays the foundations for the theory of analytic functions of many variables, builds the theory of divisibility of power series.
Consider the development of the theory of analytic functions in Russia. Russian mathematicians of the XIX 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 for whom 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). About 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 the Cauchy formula for the deduction of a function with respect to the n-th order pole. He outlined the applications of residue theory and Cauchy's 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 defined initially for real x as real and
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 20th century. fundamental research in the theory of functions of a complex variable (F. Klein, A. Poincaré, P. Kebe) consisted in the gradual elucidation of the fact that Lobachevsky's geometry 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 "Theory of integral residues with some applications" (St. Petersburg, 1868). From the autumn 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 a power series (Lagrange series) and, in particular, to the expansion of analytic functions into continued fractions, as well as to the Legendre polynomials. In this paper, the famous theorem on the behavior of an analytic function in a neighborhood of an essential singular point is formulated and proved. In Sokhotsky's doctoral dissertation
(1873) for the first time the concept of an integral of Cauchy type is introduced in an expanded form: *r/ ^ & _ where
a and b are two arbitrary complex numbers. The integral is supposed to be taken along some curve (“trajectory”) connecting a and b. In this work, a number of theorems are proved.
A huge role in the history of analytic functions was played by the works of N.E. Zhukovsky and S.A. Chaplygin, who opened up a boundless area of its applications in aero- and hydromechanics.
Speaking about the development of the theory of analytic functions, one cannot fail to mention the studies of S.V. Kovalevskaya, although their main meaning lies outside 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 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, Riemann's geometric and physical ideas, the analytical direction of Weierstrass), now the differences and the controversies associated with them are being overcome, appearing and rapidly growing the number of works in which a synthesis of ideas and methods is carried out. One of the basic concepts on which the connection and correspondence between 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 an extensive complex of disciplines: the geometric theory of functions based on the theory of conformal mappings and Riemann surfaces. We received an integral 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 has been developed. The analytic theory of differential equations and the analytic theory of numbers adjoined this complex. The theory of analytic functions established and strengthened links with other mathematical disciplines.
The wealth of interrelations between the TFCT and algebra, geometry and other sciences, the creation of the systematic foundations of the science of the TFCT itself, and its great practical significance contributed to the formation of the TFCT as an academic subject. However, simultaneously with the completion of the formation of the foundations, new ideas were introduced into the theory of analytic functions, significantly changing its composition, nature and goals. Monographs appear containing a systematic exposition of the theory of analytic functions in a style close to axiomatic and also having educational purposes. Apparently, the significance of the results on the TFCT, obtained by scientists of the period under review, prompted them to popularize the TFCT in the form of lecturing and publishing monographic studies in a teaching perspective. It can be concluded that the TFCT appeared as a learning
subject. In 1856, Ch. Briot and T. Bouquet published a small memoir "Investigation of the Functions of an Imaginary Variable", which is essentially the first textbook. General concepts in the theory of the function of a complex variable began to be worked out in lectures. Since 1856, K. Weiersht-rass 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's lectures served for many years as a prototype for textbooks on the theory of functions of a complex variable, which began to appear quite often since then. It was in his lectures that the modern standard of rigor in mathematical analysis was basically built and the structure that became traditional was singled out.
REFERENCES
1. Andronov I.K. Mathematics of real and complex numbers. M.: Education, 1975.
2. Klein F. Lectures on the development of mathematics in the XIX century. M.: ONTI, 1937. Part 1.
3. Lavrentiev 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 19th century. Geometry. Theory of Analytic Functions / ed. A. N. Kolmogorova and A. P. Yushkevich. Moscow: Nauka, 1981.
6. Mathematical Encyclopedia / Chap. ed. I. M. Vinogradov. M.: Soviet encyclopedia, 1977. T. 1.
7. Mathematical Encyclopedia / Chap. ed. I. M. Vinogradov. M.: Soviet Encyclopedia, 1979. Vol. 2.
8. Young V.N. Fundamentals of the doctrine of number in the 18th and early 19th centuries. Moscow: Uchpedgiz, 1963.
9. Rybnikov K.A. History of mathematics. M.: Publishing House of Moscow State University, 1963. Part 2.
NOT. Lyakhova TOUCHING OF PLANE CURVES
The question of the tangency of plane curves, in the case when the abscissas of the common points are found from an equation of the form Рп x = 0, where Р 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 also shown.
If the curves that are graphs of the functions y \u003d f (x) and y \u003d cp x have a common point
M() x0; v0 , i.e. y0 \u003d f x0 \u003d cp x0 and tangents to the indicated curves drawn at the point M () x0; v0 do not coincide, then we say that the curves y = fix) and y - cp x intersect at the point Mo xo;
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 (original function) to be replaced by a function of a complex variable (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 method of integral transformations is the Laplace Transform.
Continuous Laplace Transform is an integral transformation that connects a function of a complex variable (image of a function) 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 semiaxis of a real variable (the function satisfies the Dirichlet conditions);
The value of the function before the initial moment is equated 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
and With
, what 
at , where c
is the abscissa of absolute convergence (some positive number).
Laplace transform (direct integral transform) from a function of a real variable is called a function of the following form (a function of a complex variable):
The function is called the original of the function, and the function is called its image. complex variable 
is called the Laplace operator, where is the angular frequency, 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 the Euler 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 typical functions. Below are some of these functions.
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 that 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 the originals of a function is valid for images of these functions.
The linearity property simplifies finding the originals of complex images, as it allows the image of a function to be represented as a sum of simple terms, and then find the originals of each presented term.
Original differentiation theorem functions
Differentiating the original function corresponds to multiplication
For non-zero initial conditions:
Under zero initial conditions (special case):
Thus, the operation of differentiating a function is replaced by an arithmetic operation in the space of function images.
Original integration theorem functions
Integration of the original function corresponds to division image of the function on the Laplace operator.
Thus, the function integration operation is replaced by an arithmetic operation in the function image space.
Similarity theorem
Changing the function argument (compression or expansion of the signal) in the time domain leads to the reverse change of the argument and the ordinate of the function image.
An increase in the pulse duration causes a compression of its spectral function and a decrease in the amplitudes of the harmonic components of the spectrum.
Delay theorem
The delay (shift, offset) of the signal by the argument of the original function by an interval leads to a change in the phase-frequency function of the spectrum (the 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, shift) of the signal in the function image argument leads to the multiplication of the original function by an exponential factor
From a practical point of view, the displacement theorem is applied in determining the images of exponential functions.
Convolution theorem
Convolution is a mathematical operation applied to two functions and that generates a third function. In other words, having the response of a certain linear system to an impulse, it is possible to calculate the response of the system to the entire signal using convolution.
Thus, the convolution of the originals of two functions can be represented as a product of images of these functions. The reconciliation theorem is used when considering transfer functions, when the response of the system (an output signal from a quadripole) is determined when a signal is applied to the input of a quadripole with an impulse 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 formula of the inverse Laplace transform(Mellin formula, Bromwich integral), which has the following form:
In this formula, the integration limits mean that the integration goes along an infinite straight line, which is parallel to the imaginary axis and intersects the real axis at the point . Given that the last expression can be rewritten as follows:
In practice, to perform the inverse Laplace transform, the function image is decomposed into the sum of simple fractions by the method of indefinite 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 displaying 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 denominator roots:
If there is a zero root in the denominator, the function is decomposed into a fraction of the type:
If there is a zero n -fold root in the denominator, the function is decomposed into a fraction of the type: 
If there is a real root in the denominator, the function is decomposed into a fraction of the type:
If there is a real n -fold root in the denominator, the function is decomposed into a fraction of the type: 
If there is an imaginary root in the denominator, the function is decomposed into a fraction of the type: 
If there are complex conjugate roots 
in the denominator, the function is decomposed into a fraction of the type: 
∙ 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 a rational fraction), then it can be decomposed into a sum of simple fractions.
∙ In a special case if the denominator of the function image is decomposed only into simple roots of the equation, then the function image can be decomposed into the sum of simple fractions as follows:
Unknown coefficients can be determined by the method of undetermined coefficients or in a simplified way using the following formula:
Function value at 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
setting functions x(t) real variable t in line function X(s) complex variable s (s = σ+ jω). Wherein x(t) called original, X(s)- image or image according to Laplace and s- Laplace transform variable. The original is indicated by a lowercase letter, and its image is indicated by a capital letter of the same name.
It is assumed that the function x(t) subjected to the Laplace transform has the following properties:
1) function x(t) is defined and piecewise differentiable on the interval )
