S complex Laplace variable definition physical meaning. Laplace transform basic definitions of the property Duhamel's formula
One way 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
And 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 arithmetic operation in the function image space.
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 function argument (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 determining 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 reconciliation theorem is used when considering transfer functions, when the reaction of the system (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 step 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 valid for representing 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 case 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.
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
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 analytical function at 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 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): Laplace transform exists if integral exists ∫ 0 ∞ | 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) 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 bound 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 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))) analytical regarding 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 ( 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 the abscissa of absolute convergence.
Note: these are sufficient conditions of existence.
- Convolution theorem
Main article: Convolution theorem
- Differentiation and integration of the original
The Laplace image of 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 using a simple relation. This is, for example, used to analyze the stability of the trajectory of a dynamic 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.)Multiplying 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 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 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 | power n (\displaystyle n)-th order | tnn! ⋅ 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 | unit function | H (t) (\displaystyle H(t)\ ) | 1 s (\displaystyle (\frac (1)(s))) | s > 0 (\displaystyle s>0) |
| 2b | unit 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 | tnn! 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) |
Notes on the table:
| ||||
Laplace transform. When a system is described by differential and integral equations, it is often convenient to use the Laplace transform to calculate them. In this case, the equations become algebraic. In this case, it can be simplified to consider the PL as the decomposition of the signal into sinusoids and exponentials. For discrete signals, the PL is called the Z-transform.
The essence of the Laplace transform
Fourier transform
Various functions of real displacements of time t PL are associated with functions of complex displacement p and vice versa.
Laplace transform
р=σ+jω – differentiation operator
The image of a function according to Laplace is a complex plane, where σ is plotted along the axis of real values, and jω is plotted along the axis of imaginary values. Moreover, each point of the plane is a complex quantity and can be represented in algebraic or polar notation. In order to find the Laplace image, the original signal is multiplied by various exponents e -σt. If σ 0, then it is a right half-plane. For each of these products, the Fourier transform is found and placed along the imaginary value axis. The upper and lower half-planes will be mirrored if the original signal is represented by a real function.
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Previously, we considered the integral Fourier transform with the kernel K(t, O = e). The Fourier transform is inconvenient in that the condition of absolute integrability of the function f(t) on the entire t axis must be satisfied. The Laplace transform allows us to free ourselves from this limitation. Definition 1. Function We will call an original any complex-valued function f(t) of a real argument t that satisfies the following conditions: 1. f(t) is continuous on the entire t axis, except for individual points at which f(t) has a discontinuity of the 1st kind, and on each finite interval of the axis * there can be only a finite number of such points; 2. the function f(t) is equal to zero for negative values of t, f(t) = 0 for 3. as t increases, the module f(t) increases no faster than the exponential function, i.e. there are numbers M > 0 and s such that for all t It is clear that if inequality (1) is true for some s = aj, then it will also be true for ANY 82 > 8]. = infs for which inequality (1) holds is called the growth index of the function f(t). Comment. In the general case, the inequality does not hold, but the estimate where e > 0 is any is valid. Thus, the function has a growth exponent 0 = For it, the inequality \t\ ^ M V* ^ 0 does not hold, but the inequality |f| ^ Mei. Condition (1) is much less restrictive than condition (*). Example 1. the function does not satisfy condition ("), but condition (1) is satisfied for any s ^ I and A/ ^ I; growth rate 5o = So this is the original function. On the other hand, the function is not the original function: it has an infinite order of growth, “o = +oo. The simplest original function is the so-called unit function. If a certain function satisfies conditions 1 and 3 of Definition 1, but does not satisfy condition 2, then the product is already an original function. For simplicity of notation, we will, as a rule, omit the factor rj(t), stipulating that all functions that we will consider are equal to zero for negative t, so that if we are talking about some function f(t), for example, o sin ty cos t, el, etc., then the following functions are always implied (Fig. 2): n=n(0 Fig. 1 Definition 2. Let f(t) be the original function. Representing the function f(t ) according to Laplace is called a function F(p) of a complex variable, defined by the formula LAPLACE TRANSFORM Basic definitions Properties Convolution of functions Multiplication theorem Finding the original from an image Using the inversion theorem of operational calculus Duhamel's formula Integrating systems of linear differential equations with constant coefficients Solving integral equations where the integral is taken over the positive semi-axis t. The function F(p) is also called the Laplace transform of the function /(/); transformation kernel K(t) p) = e~pt. We will write down the fact that the function has F(p) as its image Example 2. Find the image of the unit function r)(t). The function is the original function with growth exponent 0 - 0. By virtue of formula (2), the image of the function rj(t) will be the function If then when the integral on the right side of the last equality will be convergent, and we will obtain so that the image of the function rj(t) will be function £. As we agreed, we will write that rj(t) = 1, and then the result obtained will be written as follows: Theorem 1. For any original function f(t) with growth index 30, the image F(p) is defined in the half-plane R e = s > s0 and is an analytical function in this half-plane (Fig. 3). Let To prove the existence of the image F(p) in the indicated half-plane, it is enough to establish that the improper integral (2) converges absolutely for a > Using (3), we obtain which proves the absolute convergence of the integral (2). At the same time, we obtained an estimate for the Laplace transform F(p) in the half-plane of convergence. Differentiating expression (2) formally under the integral sign with respect to p, we find. The existence of integral (5) is established in the same way as the existence of integral (2) was established. Applying integration by parts for F"(p), we obtain an estimate which implies the absolute convergence of integral (5). (The non-integral term,0.,- at t +oo has a limit equal to zero). In any half-plane Rep ^ sj > "o integral (5) converges uniformly with respect to p, since it is dominated by a convergent integral independent of p. Consequently, differentiation with respect to p is legal and equality (5) is valid. Since the derivative F"(p) exists, the Laplace transform F(p) is everywhere in the half-plane Rep = 5 > 5о is an analytical function. The Corollary follows from inequality (4). If the value of p tends to infinity so that Re p = s increases without limit, then Example 3. Let us also find an image of the function of any complex number. The exponent of the function /(()) is equal to a. 4 Assuming Rep = i > a, we obtain Thus, For a = 0 we again obtain the formula Let us pay attention to the fact that the image of the function eat is an analytical function of the argument p not only in the half-plane Rep > a, but also at all points p, except for the point p = a, where this image has a simple pole. In the future, we will more than once encounter a similar situation when the image F(p) will be an analytical function in the entire plane of the complex variable p, for except for isolated singular points. There is no contradiction with Theorem 1. The latter only states that in the half-plane Rep > o the function F(p) has no singular points: all of them turn out to lie either to the left of the line Rep = so, or on this line itself. Don't notice. In operational calculus, the Heaviside representation of the function f(f) is sometimes used, which is defined by the equality and differs from the Laplace representation by the factor p. §2. Properties of the Laplace transform In what follows, we will denote the original functions, and their Laplace images. From the definition of an image it follows that if Theorem 2 (unity). £biw dee continuous functions) have the same image, then they are identically equal. Teopewa 3 (p'ieiost* Laplace's transformation). If the functions are originals, then for any complex constants α The validity of the statement follows from the linearity property of the integral that defines the image: , are the growth indicators of the functions, respectively). Based on this property, we obtain Similarly, we find that and, further, Theorem 4 (similarities). If f(t) is the original function and F(p) is its Laplace image, then for any constant a > O. Setting at = m, we have Using this theorem, from formulas (5) and (6) we obtain Theorem 5 ( on differentiation of the original). Let be the original function with the image F(p) and let be also the original functions, and where is the growth index of the function Then and in general Here we mean the right limit value Let. Let's find the image We have Integrating by parts, we obtain The out-of-integral term on the right side of (10) vanishes as k. For Rc р = s > з we have the substitution t = Odets -/(0). The second term on the right in (10) is equal to pF(p). Thus, relation (10) takes the form and formula (8) is proven. In particular, if To find the image f(n\t) we write from where, integrating n times by parts, we obtain Example 4. Using the theorem on differentiation of the original, find the image of the function f(t) = sin2 t. Let Therefore, Theorem 5 establishes a remarkable property of the integral Laplace transform: it (like the Fourier transform) transforms the operation of differentiation into the algebraic operation of multiplication by p. Inclusion formula. If they are original functions, then In fact, By virtue of the corollary to Theorem 1, every image tends to zero at. This means that the inclusion formula follows (Theorem 6 (on differentiation of an image). Differentiation of an image is reduced to multiplication by the original. Since the function F(p) in the half-plane so is analytic, it can be differentiated with respect to p. We have The latter just means that Example 5. Using Theorem 6, find the image of the function 4 As is known, Hence (Applying Theorem 6 again, we find, in general, Theorem 7 (integration of the original). Integration of the original is reduced to dividing the image by Let It is not difficult to check, that if there is an original function, then it will be an original function, and Let. Due to so On the other hand, whence F= The latter is equivalent to the proved relation (13). Example 6. Find the image of the function M In this case, so that Therefore, Theorem 8 (image integration).If the integral converges, then it serves as an image of the function ^: LAPLACE TRANSFORM Basic definitions Properties Convolution of functions Multiplication theorem Finding the original from an image Using the inversion theorem of operational calculus Duhamel's formula Integrating systems of linear differential equations with constant coefficients Solution integral equations Indeed, Assuming that the path of integration lies in the half-plane so, we can change the order of integration. The last equality means that it is an image of a function Example 7. Find an image of a function M As is known, . Therefore, since we assume that we get £ = 0, when. Therefore, relation (16) takes the form Example. Find the image of the function f(t), specified graphically (Fig. 5). Let us write the expression for the function f(t) in the following form: This expression can be obtained as follows. Consider the function and subtract the function from it. The difference will be equal to one for. To the resulting difference we add the function. As a result, we obtain the function f(t) (Fig. 6 c), so that from here, using the delay theorem, we find Theorem 10 (displacement). then for any complex number po In fact, the Theorem makes it possible to use known images of functions to find images of the same functions multiplied by an exponential function, for example, 2.1. Function folding. Multiplication theorem Let the functions f(t) be defined and continuous for all t. The convolution of these functions is called new feature on t, defined by equality (if this integral exists). For original functions, the operation convolve is always feasible, and (17) 4 In fact, the product of original functions as a function of m is a finite function, i.e. vanishes outside some finite interval (in this case, outside the segment. For finite continuous functions, the convolution operation is feasible, and we obtain the formula It is not difficult to verify that the convolution operation is commutative, Theorem 11 (multiplication). If, then the convolution t) has an image It is not difficult to verify that that the convolution (of the original functions is the original function with the growth exponent » where, are the growth exponents of the functions, respectively. Let us find the image of the convolution. Using what we have. Changing the order of integration in the integral on the right (such an operation is legal) and applying the retardation theorem, we obtain Thus Thus, from (18) and (19) we find that the multiplication of images corresponds to the convolution of the originals, Prter 9. Find the image of the function A function V(0) is the convolution of functions. By virtue of the multiplication theorem Problem. Let the function /(ξ) be periodic with period T , is the original function. Show that its Laplace image F(p) is given by formula 3. Finding the original from the image The problem is posed as follows: given the function F(p), we need to find the function /(<)>whose image is F(p). Let us formulate conditions sufficient for the function F(p) of a complex variable p to serve as an image. Theorem 12. If a function F(p) analytic in the half-plane so 1) tends to zero as in any half-plane R s0 uniformly with respect to arg p; 2) the integral converges absolutely, then F(p) is the image of some original function Problem. Can the function F(p) = serve as an image of some original function? We will indicate some ways to find the original from an image. 3.1. Finding the original using image tables First of all, it is worth bringing the function F(p) to a simpler, “tabular” form. For example, in the case when F(p) is a fractional rational function of the argument p, it is decomposed into elementary fractions and the appropriate properties of the Laplace transform are used. Example 1. Find the original for We write the function F(p) in the form Using the displacement theorem and the linearity property of the Laplace transform, we obtain Example 2. Find the original for the function 4 We write F(p) in the form Hence 3.2. Using the inversion theorem and its corollaries Theorem 13 (inversion). If the function fit) is the original function with growth exponent s0 and F(p) is its image, then at any point of continuity of the function f(t) the relation is satisfied where the integral is taken along any straight line and is understood in the sense of the principal value, i.e. as Formula (1) is called the Laplace transform inversion formula, or Mellin's formula. Indeed, let, for example, f(t) be piecewise smooth on each finite segment)
