Laplace transform

The Laplace transform, named after Pierre -Simon Laplace, is a one-sided integral transformation that a given function f from the real time domain into a function F in the complex spectral (frequency range, image area) transferred. This function F is called the spectral function or Laplace transform. The Fourier transform the Laplace transform has some similarities. So there is the Laplace transform, also an inverse transform, which is also called Bromwich integral.

• 7.1 Limit Theorems
• 7.2 uniqueness
• 7.3 Relationship to the Fourier transform

• 8.1 analyticity
• 8.2 Conjugate Symmetry
• 8.3 Finite Laplace transform

General

The Laplace transform and its inversion, methods for solving problems of mathematical physics and theoretical electrical engineering, which are described mathematically by linear initial and boundary value problems. Laplace transformation belongs to the class of functional transformations to the specific integral transforms, and is closely related to the Fourier transform. The motivation to develop further the Fourier transform to the Laplace transform, is located in the restricted class of functions, for which there is the Fourier integral in the Fourier transform.

To ensure the convergence for a wider range of functions in the Laplace transform, one extends the integration of the Fourier integral by a factor going for and against 0 and so for sufficiently large t the convergence guarantees. So that the frequency parameters of the Fourier transform is the complex frequency parameter (this is the usual designation in electrical engineering for the imaginary unit: while the available in the mathematics for common electric current ). In contrast to the two-sided Laplace and the Fourier transform, the ordinary one-sided Laplace transform is defined only for positive values ​​of t ≥ 0. This restriction is therefore permissible, as in the context of systems theory and application in the field of physics and technology only real existing causal systems play a role. In the theoretical study of non-causal systems, the two-sided Laplace transform is needed.

The Laplace transform is original functions from a real variable on screen functions of a complex variable. When existence of the Laplace transform corresponds to the differentiation and integration in the real original area of ​​a simple algebraic operation in the image domain, which explains the practical significance. In many initial and boundary value problems of the time domain plays the role of the real original range and the frequency range or spectrum that of the complex image area.

The investigation of the image function often give much better physical insight into the behavior of linear systems with respect to studies in the time domain. In particular, the resonance behavior of physical systems can be described in the frequency domain easier. Due to the better convergence compared to the Fourier transform, for example, transfer functions can be then analyzed when a linear system behaves unstable.

Wherein the discrete-time systems is in order to avoid the periodicity in the s- plane ( this is due to the discrete time sample values ​​in the discrete Laplace transformation ) is carried out a conformal, non-linear mapping to the so-called z-plane, resulting in the Z transformation leads.

History

The first references to the idea of the Laplace transform can be found already in the work of the Basel mathematician and physicist Leonhard Euler (1707-1783, Institutiones calculi integrali, vol. 2, 1768). Is called the Laplace transform of the French mathematician and astronomer Pierre- Simon Laplace (1749-1827), who in 1782 introduced the transformation in the context of probability studies. In fact, the Hungarian mathematician JOZEPH Miksa Petzval (1807-1891), the first, who systematically studied, whereas Laplace it applied only to solve his problems. However, the work of Petzval was never addressed, among other things, because it one of his students had unjustly accused of plagiarism in Laplace.

About a hundred years later, the British electrical engineer and physicist Oliver Heaviside (1850-1925) turned to the operational calculus it has found by trial and error for solving differential equations in theoretical electrical engineering. The German mathematician Gustav Doetsch (1892-1977) replaced it with the Laplace transform, developed the mathematical foundations and led the Laplace transform of a broad application to the solution of many problems of mathematical physics and theoretical electrical engineering to which by linear initial and boundary value problems are described. At least since the early 60s of the 20th century, see Theory and Application of the Laplace transform in textbooks and curricula of theoretical electrical engineering, and especially in books on ordinary and partial differential equations.

A purely algebraic reasoning and extension of the operational calculus of Heaviside led by the Polish mathematician January Mikusinski (1913-1987), without using the Laplace transform. This extension also covers functions that do not have a Laplace transform, and provides, for example, a simple exact reason for the delta function without recourse to distributions.

Definition

Let be a function. The Laplace transform of being carried

Defined, in that the integral exists. There is an ( improper ) integral parameter with the parameter. The exponential function is the core of the Laplace transform. The function is called Laplace transform of the function.

Existence

A function is said of exponential order if there are constants, with, and a place with so

Applies. Sometimes the more stringent condition is used instead.

If is of exponential order and thus the sizes are given from the above equation, and if in addition

Holds, then there exists in the half-plane, the Laplace integral.

The condition is already satisfied if the function is continuous in piecemeal.

Examples of functions whose Laplace integral exists are listed in the correspondence tables below.

The above conditions are sufficient for the existence of the Laplace integral. If they are not fulfilled, one needs to investigate further.

A few examples:

• The function is of exponential order and also does not have a Laplace transform.

Laplace reverse transformation

To the Laplace transform, it is also an inverse transformation, that is an operator of a given spectral function assigns the appropriate time. This integral operator is also called the Bromwich integral and is named after the mathematician and physicist Thomas John I'Anson Bromwich.

Statement

The time function by the inversion formula

Be determined from the spectral function, here is the of convergence of.

Example

Known back transformations are summarized in the literature in correspondence tables. In practice, therefore, the spectral function usually has to be attributed only to those tabulated cases, eg by partial fractions.

As an example, the inverse transformation of rational functions, let us consider: For the spectral function

Can be combined with the ( tabulated, calculated here as examples ) Correspondence

Specify the inverse transform directly as

In the case of complex conjugate poles of trigonometric simplifications by using identities are possible.

Important applications

Generally there is the Laplace transform to the solution of linear differential equations or systems of differential equations with constant coefficients. The advantage here is the algebraization: derivations arise in the image domain as the product of the Laplace transforms of the original function and the Laplace factor s This causes

• Ordinary differential equations in the original domain to algebraic equations in the image area,

• Partial differential equations with n independent variables in the original domain to partial (or ordinary ) differential equations with nl independent variables in the image area,

• And integral equations of convolution type in the original domain to algebraic equations in the image area

Are mapped. The solutions of the transformed problems can be much easier than in the original work area in the image field. In special cases, linear differential equations with polynomial coefficients can be solved.

Very efficiently if the Laplace transform is used to solve initial value problems, since the initial values ​​enter into the equation image. Transforming the equation into the spectral domain, triggers the algebraic equation thus obtained, and transforming the solution back into the time domain. It should be pointed out once again that the result obtained solely statements for the period from t = 0 yields, since the Laplace transform is determined by integrating from t = 0.

The disadvantage is the most complicated in general inverse transformation.

In mechanical and electrical engineering, especially in control engineering, the Laplace transform plays an important role, mainly due to the convolution theorem. Since the behavior of the system output can be represented as a product of the input function and the system 's own, independent of the respective excitation transfer function in the spectral domain, many system properties can be determined (which can, in turn, obtained by simple combinations of elementary transfer functions ) by examining the transfer function without a explicit solution to the system of differential equations, for example by back-transformation, determine. Elegant is possible by, for example, the stability analysis and analysis of the vibration characteristics (attenuation), the speed of both control systems as well as closed-loop systems. Since the transfer function in the Laplace domain for merges into a transfer function in the Fourier domain, can be finally and graphical representations of the transmission behavior, ie gain amplitude and phase frequency responses ( Bode plots ).

Properties

Limit theorems

In particular, each Laplace transform tends to when to seek. The first limit theorem applies only if other than a simple pole at has no further singularities in the half-plane.

Unambiguity

When there are two time functions and the conditions:

• And are piecewise continuous
• And are of exponential order for
• The Laplace transform and exist
• In the region of convergence

Then everywhere and are steadily.

Comparisons: uniqueness theorem of Lerch

Relation to the Fourier transform

The Laplace transform is a similar integral transform such as Fourier transform. If, with real, so the special case arises

This integral transform is sometimes called unilateral Fourier transform.

Analytical properties

Analyticity

The Laplace transform is a result of the existence of their derivatives with respect to the complex frequency in the image area

Inside the convergence half-plane as many times differentiable complex, which is called analytic (or regular or holomorphic ). Thus it can be studied with the methods of function theory. The function can be analytically continued into the left half-plane, but not mandatory. Such analytic continuation can then no longer be written as Laplace.

Conjugate symmetry

Another important property of the Laplace transform of real time functions, the complex conjugate symmetry in the image area

Or separated into real and imaginary parts

Where the overline denotes the complex conjugate size. Due to this property, it is sufficient to study the function image in the upper half plane.

Finite Laplace transform

The Laplace transform is a finite time function

To an entire function from. This means that the image function

Is analytic in the whole complex frequency plane, ie has no singularities.

Physical dimension

For applications of the Laplace transform is the dimension of the Laplace transform

Of interest. The complex has the frequency dimension. The term in the integrand is therefore dimensionless. By integrating over the time domain, the dimension of time function is multiplied by the dimension of the time derivative:

For example, has the Laplace transform of a current

The dimension of a charge As = C.

Correspondence tables

Main Features

Correspondence table

For the original function.

Example

In the following the solution of the initial value problem of an ordinary differential equation of first order with constant coefficients is presented with the help of the Laplace transform:

With. Taking advantage of the linearity of Laplace transforms and behavior for the discharge in the source area (see Table of the general properties of ) the transform is given by

With. The transformation back to the source area is in the above correspondence table ( see exponential ),

The above differential equation thus describes the simple growth and decline processes and therefore can be found in many areas, including in natural sciences, economics and social sciences.