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Encyclopedia > Laplace transform

In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original) with a real argument t (t ≥ 0) that transforms it to a function F(s) (image) with a complex argument s. This transformation is essentially bijective for the majority of practical uses; the respective pairs of f(t) and F(s) are matched in tables. The Laplace transform has the useful property that many relationships and operations over the originals f(t) correspond to simpler relationships and operations over the images F(s)[1]. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... A bijective function. ...


The Laplace transform has many important applications in mathematics, physics, optics, electrical engineering, control engineering, signal processing, and probability theory. In mathematics, it is used for solving differential and integral equations. In physics, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In this analysis, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, or radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For the book by Sir Isaac Newton, see Opticks. ... Electrical Engineers design power systems. ... Control engineering is the engineering discipline that focuses on the mathematical modelling systems of a diverse nature, analysing their dynamic behaviour, and using control theory to make a controller that will cause the systems to behave in a desired manner. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... Linear time invarient systems are called as lti systems it should satisify both linearity and time invarient quality. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... An electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... An undamped spring-mass system is a simple harmonic oscillator. ... An optical instrument either processes light waves to enhance an image for viewing, or analyzes light waves (or photons) to determine one of a number of characteristic properties. ... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... It has been suggested that this article or section be merged into Angular velocity. ... In mathematics and physics, the radian is a unit of angle measure. ...


The Laplace transform is named in honor of mathematician and astronomer Pierre-Simon Laplace, who used the transform in his work on probability theory. Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Galileo is often referred to as the Father of Modern Astronomy. ... Pierre-Simon, marquis de Laplace (March 23, 1749 - March 5, 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ...

Contents

History

From 1744, Leonhard Euler investigated integrals of the form: // Events The third French and Indian War, known as King Georges War, breaks out at Port Royal, Nova Scotia The First Saudi State founded by Mohammed Ibn Saud Prague occupied by Prussian armies Ongoing events War of the Austrian Succession (1740-1748) Births January 10 - Thomas Mifflin, fifth President... Euler redirects here. ...

— as solutions of differential equations but did not pursue the matter very far.[2] Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form: Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

— which some modern historians have interpreted within modern Laplace transform theory.[3][4]


These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.[5] However, in 1785, Laplace took the critical step forward when, rather than just look for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form: 1782 was a common year starting on Tuesday (see link for calendar). ... 1785 was a common year starting on Saturday (see link for calendar). ...

— akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.[6] In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...


Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space as the solutions were periodic. In 1809, Laplace applied the eponymous transform to find solutions that diffused indefinitely in space.[7] Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... Year 1809 (MDCCCIX) was a common year starting on Sunday (link will display the full calendar). ...


Formal definition

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by: This article is about functions in mathematics. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

The lower limit of 0 is short notation to mean

and assures the inclusion of the entire Dirac delta function δ(t) at 0 if there is such an impulse in f(t) at 0. The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...


The parameter s is in general complex: In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

This integral transform has a number of properties that make it useful for analyzing linear dynamic systems. The most significant advantage is that differentiation and integration become multiplication and division, respectively, by s. (This is similar to the way that logarithms change an operation of multiplication of numbers to addition of their logarithms.) This changes integral equations and differential equations to polynomial equations, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts back to the time domain. In mathematics, an integral transform is any transform T of the following form: The input of this transform is a function f, and the output is another function Tf. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... For other uses, see Derivative (disambiguation). ... This article is about the concept of integrals in calculus. ... Look up logarithm in Wiktionary, the free dictionary. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...


Bilateral Laplace transform

Main article: Two-sided Laplace transform

When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is normally intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform or two-sided Laplace transform by extending the limits of integration to be the entire real axis. If that is done the common unilateral transform simply becomes a special case of the bilateral transform where the definition of the function being transformed is multiplied by the Heaviside step function. In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. ... In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...


The bilateral Laplace transform is defined as follows:

Inverse Laplace transform

For more details on this topic, see Inverse Laplace transform.

The inverse Laplace transform is given by the following complex integral, which is known by various names (the Bromwich integral, the Fourier-Mellin integral, and Mellin's inverse formula): In mathematics, the Bromwich integral or inverse Laplace transform of F(s) is the function f(t) which has the property where is the Laplace transform. ... In mathematics, the Bromwich integral or inverse Laplace transform of F(s) is the function f(t) which has the property where is the Laplace transform. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

where γ is a real number so that the contour path of integration is in the region of convergence of F(s) normally requiring γ > Re(sp) for every singularity sp of F(s) and i2 = −1. If all singularities are in the left half-plane, that is Re(sp) < 0 for every sp, then γ can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...


An alternative formula for the inverse Laplace transform is given by Post's inversion formula. Posts inversion formula for Laplace transforms, named after Emil Post, is a simple-looking but usually impractical formula for evaluating an inverse Laplace transform. ...


Region of convergence

The Laplace transform F(s) typically exists for all complex numbers such that Re{s} > a, where a is a real constant which depends on the growth behavior of f(t), whereas the two-sided transform is defined in a range a < Re{s} < b. The subset of values of s for which the Laplace transform exists is called the region of convergence (ROC) or the domain of convergence. In the two-sided case, it is sometimes called the strip of convergence.


The integral defining the Laplace transform of a function may fail to exist for various reasons. For example, when the function has infinite discontinuities in the interval of integration, or when it increases so rapidly that exp(-pt) cannot dampen it sufficiently for convergence on the interval to take place. There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken, other than to say the defining integral converges. It is however easy to give theorems on cases where it may or may not be taken.


Properties and theorems

Given the functions f(t) and g(t), and their respective Laplace transforms F(s) and G(s):

the following table is a list of properties of unilateral Laplace transform:

Properties of the unilateral Laplace transform
Time domain Frequency domain Comment
Linearity Can be proved using basic rules of integration.
Frequency differentiation
Frequency differentiation More general form
Differentiation Write the exact integral form of the given function, and add another integral to complement the former to deduce the sum to indefinite integration of a differential. Next few steps are simple.
Second Differentiation Apply the Differentiation property to f'(t).
General Differentiation Follow the process briefed for the Second Differentiation.
Frequency integration
Integration u(t) is the Heaviside step function. Note u(t) * f(t) is the convolution of u(t) and f(t), not multiplication.
Scaling
Frequency shifting
Time shifting u(t) is the Heaviside step function
Convolution
Periodic Function f(t) is a periodic function of period T so that
  • Initial value theorem:
  • Final value theorem:
, all poles in left-hand plane.
The final value theorem is useful because it gives the long-term behaviour without having to perform partial fraction decompositions or other difficult algebra. If a function's poles are in the right hand plane (e.g. et or sin(t)) the behaviour of this formula is undefined.

The word linear comes from the Latin word linearis, which means created by lines. ... For other uses, see Frequency (disambiguation). ... For other uses, see Derivative (disambiguation). ... For other uses, see Derivative (disambiguation). ... For other uses, see Derivative (disambiguation). ... For other uses, see Frequency (disambiguation). ... This article is about the concept of integrals in calculus. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In algebra, the partial fraction decomposition or (partial fraction expansion) is used to reduce the degree of either the numerator or the denominator of a rational function. ...

Proof of the Laplace transform of a function's derivative

It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. This can be derived from the basic expression for a Laplace transform as follows:

(by parts)

yielding

and in the bilateral case, we have

Relationship to other transforms

Fourier transform

The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument s = iω or s = 2πfi: In mathematics, the continuous Fourier transform is a certain linear operator that maps functions to other functions. ...

Note that this expression excludes the scaling factor , which is often included in definitions of the Fourier transform.


This relationship between the Laplace and Fourier transforms is often used to determine the frequency spectrum of a signal or dynamic system. Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ...


Mellin transform

The Mellin transform and its inverse are related to the two-sided Laplace transform by a simple change of variables. If in the Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. ...

we set θ = e-t we get a two-sided Laplace transform.


Z-transform

The Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

where is the sampling period (in units of time e.g., seconds) and is the sampling rate (in samples per second or hertz)

Let The Nyquist-Shannon sampling theorem is the fundamental theorem in the field of information theory, in particular telecommunications. ... The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ... A sample refers to a value or set of values at a point in time and/or space. ... This article is about the SI unit of frequency. ...

be a sampling impulse train (also called a Dirac comb) and In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions for some given period T. Some authors, notably Bracewell, refer to it as the Shah function (probably because its graph resembles the shape of the cyrillic letter sha Ш). From the orthogonality of the Fourier series...

be the continuous-time representation of the sampled .

are the discrete samples of .

The Laplace transform of the sampled signal is

This is precisely the definition of the Z-transform of the discrete function In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

with the substitution of .


Comparing the last two equations, we find the relationship between the Z-transform and the Laplace transform of the sampled signal:

The similarity between the Z and Laplace transforms is expanded upon in the theory of time scale calculus. In mathematics, time scale calculus is a unification of the theory of difference equations and standard calculus. ...


Borel transform

The integral form of the Borel transform is identical to the Laplace transform; indeed, these are sometimes mistakenly assumed to be synonyms. The generalized Borel transform generalizes the Laplace transform for functions not of exponential type. In mathematics, in the area of complex analysis, Nachbins theorem is commonly used to establish a bound on the growth rates for an analytic function. ... In mathematics, in the area of complex analysis, Nachbins theorem is commonly used to establish a bound on the growth rates for an analytic function. ... In mathematics, in the area of complex analysis, Nachbins theorem is commonly used to establish a bound on the growth rates for an analytic function. ...


Fundamental relationships

Since an ordinary Laplace transform can be written as a special case of a two-sided transform, and since the two-sided transform can be written as the sum of two one-sided transforms, the theory of the Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same subject. However, a different point of view and different characteristic problems are associated with each of these four major integral transforms.


Table of selected Laplace transforms

The following table provides Laplace transforms for many common functions of a single variable. For definitions and explanations, see the Explanatory Notes at the end of the table.


Because the Laplace transform is a linear operator:

  • The Laplace transform of a sum is the sum of Laplace transforms of each term.
  • The Laplace transform of a multiple of a function, is that multiple times the Laplace transformation of that function.

The unilateral Laplace transform is only valid when t is non-negative, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t).

ID Function Time domain
Laplace s-domain
Region of convergence
for causal systems
1 ideal delay
1a unit impulse
2 delayed nth power
with frequency shift
2a nth power
( for integer n )
2a.1 qth power
( for real q )
2a.2 unit step
2b delayed unit step
2c ramp
2d nth power with frequency shift
2d.1 exponential decay
3 exponential approach
4 sine
5 cosine
6 hyperbolic sine
7 hyperbolic cosine
8 Exponentially-decaying
sine wave
9 Exponentially-decaying
cosine wave
10 nth root
11 natural logarithm
12 Bessel function
of the first kind,
of order n

13 Modified Bessel function
of the first kind,
of order n
14 Bessel function
of the second kind,
of order 0
15 Modified Bessel function
of the second kind,
of order 0
   
16 Error function
Explanatory notes:

A causal system is a system that depends only on the current and previous inputs. ... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... The ramp function is an elementary unary real function, easily computable as the mean of its independent variable and its absolute value. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred... In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: x2 for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred... Plot of the error function In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ... The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Γ) is an extension of the factorial function to real and complex numbers. ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... It has been suggested that this article or section be merged into Angular velocity. ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... Please refer to Real vs. ... Not to be confused with Natural number. ... A causal system is a system that depends only on the current and previous inputs. ... The Impulse response from a simple audio system. ... An acausal system is a system that depends on both the past and the future. ... Causality describes the relationship between causes and effects, and is fundamental to all natural science, especially physics. ...

s-Domain equivalent circuits and impedances

The Laplace transform is often used in circuit analysis, and simple conversions to the s-Domain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances. Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... Sine waves are commonly found in physics and engineering, such as in the study of alternating current (AC) circuits. ...


Here is a summary of equivalents:

Note that the resistor is exactly the same in the time domain and the s-Domain. The sources are put in if there are initial conditions on the circuit elements. For example, if a capacitor has an initial voltage across it, or if the inductor has an initial current through it, the sources inserted in the s-Domain account for that. Image File history File links S-domain_circuit_equivalents. ...


The equivalents for current and voltage sources are simply derived from the transformations in the table above.


Examples: How to apply the properties and theorems

The Laplace transform is used frequently in engineering and physics; the output of a linear dynamic system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see control theory. Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ... The Impulse response from a simple audio system. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics, multiplication is an elementary arithmetic operation. ... For control theory in psychology and sociology, see control theory (sociology). ...


The Laplace transform can also be used to solve differential equations and is used extensively in electrical engineering. The method of using the Laplace Transform to solve differential equations was developed by the English electrical engineer Oliver Heaviside. The use of Laplace transform makes it much easier to solve linear differential equations with given initial conditions. ... Electrical Engineers design power systems. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Oliver Heaviside (May 18, 1850 – February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and...

The following examples, derived from applications in physics and engineering, will use SI units of measure. SI is based on meters for distance, kilograms for mass, seconds for time, and amperes for electric current.

A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... metre or meter, see meter (disambiguation) The metre is the basic unit of length in the International System of Units. ... The international prototype, made of platinum-iridium, which is kept at the BIPM under conditions specified by the 1st CGPM in 1889. ... This article is about the unit of time. ... For other uses, see Ampere (disambiguation). ...

Example #1: Solving a differential equation

The following example is based on concepts from nuclear physics.

Consider the following first-order, linear differential equation: This box:      Nuclear physics is the branch of physics concerned with the nucleus of the atom. ...

This equation is the fundamental relationship describing radioactive decay, where Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. ...

represents the number of undecayed atoms remaining in a sample of a radioactive isotope at time t (in seconds), and is the decay constant. For other uses, see Isotope (disambiguation). ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...


We can use the Laplace transform to solve this equation.


Rearranging the equation to one side, we have

Next, we take the Laplace transform of both sides of the equation:

where

and

Solving, we find

Finally, we take the inverse Laplace transform to find the general solution

which is indeed the correct form for radioactive decay.


Example #2: Deriving the complex impedance for a capacitor

This example is based on the principles of electrical circuit theory.

The constitutive relation governing the dynamic behavior of a capacitor is the following differential equation: An electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... See Capacitor (component) for a discussion of specific types. ...

where C is the capacitance (in farads) of the capacitor, i = i(t) is the electrical current (in amperes) flowing through the capacitor as a function of time, and v = v(t) is the voltage (in volts) across the terminals of the capacitor, also as a function of time. The farad (symbol F) is the SI unit of capacitance (named after Michael Faraday). ... In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ... In physics, the ampere (symbol: A, often informally abbreviated to amp) is the SI base unit used to measure electrical currents. ... Electric potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... The volt is the SI derived unit for electric potential and voltage (derived from the ampere and watt). ...


Taking the Laplace transform of this equation, we obtain

where

and

Solving for V(s) we have

The definition of the complex impedance Z (in ohms) is the ratio of the complex voltage V divided by the complex current I while holding the initial state Vo at zero: In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... The ohm (symbol: Ω) is the SI unit of electric resistance. ...

Using this definition and the previous equation, we find:

which is the correct expression for the complex impedance of a capacitor.


Example #3: Finding the transfer function from the impulse response

Relationship between the time domain and the frequency domain. Note the * in the time domain, denoting convolution.
This example is based on concepts from signal processing, and describes the dynamic behavior of a damped harmonic oscillator. See also RLC circuit.

Consider a linear time-invariant system with impulse response Image File history File links Download high resolution version (1728x1296, 97 KB) Summary I created this image myself in Mocrosoft PowerPoint and saved it as a PNG file. ... Image File history File links Download high resolution version (1728x1296, 97 KB) Summary I created this image myself in Mocrosoft PowerPoint and saved it as a PNG file. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... A harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement : where is a positive constant. ... An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ... The Impulse response from a simple audio system. ...

such that

where t is the time (in seconds), and

is the phase delay (in radians). This article is about a portion of a periodic process. ... In mathematics and physics, the radian is a unit of angle measure. ...


Suppose that we want to find the transfer function of the system. We begin by noting that A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

where

is the time delay of the system (in seconds), and is the Heaviside step function. The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...


The transfer function is simply the Laplace transform of the impulse response:

where

is the (undamped) natural frequency or resonance of the system (in radians per second). This article is about resonance in physics. ... This article is about resonance in physics. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ...


Example #4: Method of partial fraction expansion

Consider a linear time-invariant system with transfer function A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

The impulse response is simply the inverse Laplace transform of this transfer function: The Impulse response from a simple audio system. ...

To evaluate this inverse transform, we begin by expanding H(s) using the method of partial fraction expansion: In algebra, the partial fraction decomposition or (partial fraction expansion) is used to reduce the degree of either the numerator or the denominator of a rational function. ...

for unknown constants P and R. To find these constants, we evaluate

and

Substituting these values into the expression for H(s), we find

Finally, using the linearity property and the known transform for exponential decay (see Item #3 in the Table of Laplace Transforms, above), we can take the inverse Laplace transform of H(s) to obtain:

which is the impulse response of the system.


Example #5: Mixing sines, cosines, and exponentials

Time function Laplace transform

Starting with the Laplace transform

we find the inverse transform by first adding and subtracting the same constant α to the numerator:

By the shift-in-frequency property, we have

Finally, using the Laplace transforms for sine and cosine (see the table, above), we have

Example #6: Phase delay

Time function Laplace transform

Starting with the Laplace transform,

we find the inverse by first rearranging terms in the fraction:

We are now able to take the inverse Laplace transform of our terms:

To simplify this answer, we must recall the trigonometric identity that In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

and apply it to our value for x(t):

We can apply similar logic to find that

See also

Pierre-Simon, marquis de Laplace (March 23, 1749 - March 5, 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... Analog signal processing is any signal processing conducted on analog signals by analog means. ... The use of Laplace transform makes it much easier to solve linear differential equations with given initial conditions. ...

References

  1. ^ Korn and Korn, Section 8.1
  2. ^ Euler (1744), (1753) and (1769)
  3. ^ Lagrange (1773)
  4. ^ Grattan-Guinness (1997) p.260
  5. ^ Grattan-Guinness (1997) p.261
  6. ^ Grattan-Guinness (1997) p.261-262
  7. ^ Grattan-Guinness (1997) p. 262-266

Bibliography

Modern

  • G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Companies; 2nd edition (June 1967). ISBN 0-0703-5370-0
  • A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
  • William McC. Siebert, Circuits, Signals, and Systems, MIT Press, Cambridge, Massachusetts, 1986. ISBN 0-262-19229-2
  • Davies, Brian, Integral transforms and their applications, Third edition, Springer, New York, 2002. ISBN 0-387-95314-0
  • Wolfgang Arendt, Charles J.K. Batty, Matthias Hieber, and Frank Neubrander. Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser Basel, 2002. ISBN-10:3764365498

Historical

  • Deakin, M. A. B. (1981). "The development of the Laplace transform". Archive for the History of the Exact Sciences 25: 343-390. 
  • — (1982). "The development of the Laplace transform". Archive for the History of the Exact Sciences 26: 351-381. 
  • Euler, L. (1744) "De constructione aequationum", Opera omnia 1st series, 22:150-161
  • — (1753) "Methodus aequationes differentiales", Opera omnia 1st series, 22:181-213
  • — (1769) Institutiones calculi integralis 2, Chs.3-5, in Opera omnia 1st series, 12
  • Grattan-Guinness, I (1997) "Laplace's integral solutions to partial differential equations", in Gillispie, C. C. Pierre Simon Laplace 1749-1827: A Life in Exact Science, Princeton: Princeton University Press, ISBN 0-691-01185-0
  • Lagrange, J. L. (1773) "Mémoire sur l'utilité de la méthode", Œuvres de Lagrange, 2:171-234

External links

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

  Results from FactBites:
 
Laplace Transform (339 words)
The Laplace transform is a close relative of the Fourier transform.
In addition, the Laplace transform is useful in determining solutions of partial differential equations -- particularly where the time or spatial domains are semi-infinite, e.g., heat flow in an infinite rod where the temperature at one end is known.
For the Laplace transform, the appropriate convolution f
PlanetMath: Laplace transform (148 words)
Notice the Laplace transform is a linear transformation.
The most popular usage of the Laplace transform is to solve initial value problems by taking the Laplace transform of both sides of an ordinary differential equation.
This is version 19 of Laplace transform, born on 2003-06-11, modified 2007-03-01.
  More results at FactBites »

 
 

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