Laplace transform

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In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace ( /ləˈplɑːs/ ), is an integral transform that converts a function of a real variable (often time) to a function of a complex variable (complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication. [1] [2] [3]

Contents

For suitable functions f, the Laplace transform is the integral

History

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform in his work on probability theory. [4] Laplace wrote extensively about the use of generating functions in Essai philosophique sur les probabilités (1814), and the integral form of the Laplace transform evolved naturally as a result. [5]

Laplace's use of generating functions was similar to what is now known as the z-transform, and he gave little attention to the continuous variable case which was discussed by Niels Henrik Abel. [6] The theory was further developed in the 19th and early 20th centuries by Mathias Lerch, [7] Oliver Heaviside, [8] and Thomas Bromwich. [9]

The current widespread use of the transform (mainly in engineering) came about during and soon after World War II, [10] replacing the earlier Heaviside operational calculus. The advantages of the Laplace transform had been emphasized by Gustav Doetsch, [11] to whom the name Laplace Transform is apparently due.

From 1744, Leonhard Euler investigated integrals of the form

as solutions of differential equations, but did not pursue the matter very far. [12] Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

which some modern historians have interpreted within modern Laplace transform theory. [13] [14] [ clarification needed ]

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. [15] However, in 1785, Laplace took the critical step forward when, rather than simply looking 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

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. [16]

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, because those solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space. [17]

Formal definition

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by

 

 

 

 

(Eq.1)

where s is a complex number frequency parameter

, with real numbers σ and ω.

An alternate notation for the Laplace transform is instead of F. [1] [3]

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞). For locally integrable functions that decay at infinity or are of exponential type, the integral can be understood to be a (proper) Lebesgue integral. However, for many applications it is necessary to regard it as a conditionally convergent improper integral at . Still more generally, the integral can be understood in a weak sense, and this is dealt with below.

One can define the Laplace transform of a finite Borel measure μ by the Lebesgue integral [18]

An important special case is where μ is a probability measure, for example, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated as though the measure came from a probability density function f. In that case, to avoid potential confusion, one often writes

where the lower limit of 0 is shorthand notation for

This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.

Bilateral Laplace transform

When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is usually 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.

The bilateral Laplace transform F(s) is defined as follows:

 

 

 

 

(Eq.2)

An alternate notation for the bilateral Laplace transform is , instead of .

Inverse Laplace transform

Two integrable functions have the same Laplace transform only if they differ on a set of Lebesgue measure zero. This means that, on the range of the transform, there is an inverse transform. In fact, besides integrable functions, the Laplace transform is a one-to-one mapping from one function space into another in many other function spaces as well, although there is usually no easy characterization of the range.

Typical function spaces in which this is true include the spaces of bounded continuous functions, the space L(0, ) , or more generally tempered distributions on (0, ). The Laplace transform is also defined and injective for suitable spaces of tempered distributions.

In these cases, the image of the Laplace transform lives in a space of analytic functions in the region of convergence. 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):

 

 

 

 

(Eq.3)

where γ is a real number so that the contour path of integration is in the region of convergence of F(s). In most applications, the contour can be closed, allowing the use of the residue theorem. An alternative formula for the inverse Laplace transform is given by Post's inversion formula. The limit here is interpreted in the weak-* topology.

In practice, it is typically more convenient to decompose a Laplace transform into known transforms of functions obtained from a table, and construct the inverse by inspection.

Probability theory

In pure and applied probability, the Laplace transform is defined as an expected value. If X is a random variable with probability density function f, then the Laplace transform of f is given by the expectation

By convention, this is referred to as the Laplace transform of the random variable X itself. Here, replacing s by t gives the moment generating function of X. The Laplace transform has applications throughout probability theory, including first passage times of stochastic processes such as Markov chains, and renewal theory.

Of particular use is the ability to recover the cumulative distribution function of a continuous random variable X, by means of the Laplace transform as follows: [19]

Region of convergence

If f is a locally integrable function (or more generally a Borel measure locally of bounded variation), then the Laplace transform F(s) of f converges provided that the limit

exists.

The Laplace transform converges absolutely if the integral

exists as a proper Lebesgue integral. The Laplace transform is usually understood as conditionally convergent, meaning that it converges in the former but not in the latter sense.

The set of values for which F(s) converges absolutely is either of the form Re(s) > a or Re(s) ≥ a, where a is an extended real constant with −∞ ≤ a ≤ ∞ (a consequence of the dominated convergence theorem). The constant a is known as the abscissa of absolute convergence, and depends on the growth behavior of f(t). [20] Analogously, the two-sided transform converges absolutely in a strip of the form a < Re(s) < b, and possibly including the lines Re(s) = a or Re(s) = b. [21] The subset of values of s for which the Laplace transform converges absolutely is called the region of absolute convergence, or the domain of absolute convergence. In the two-sided case, it is sometimes called the strip of absolute convergence. The Laplace transform is analytic in the region of absolute convergence: this is a consequence of Fubini's theorem and Morera's theorem.

Similarly, the set of values for which F(s) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the region of convergence (ROC). If the Laplace transform converges (conditionally) at s = s0, then it automatically converges for all s with Re(s) > Re(s0). Therefore, the region of convergence is a half-plane of the form Re(s) > a, possibly including some points of the boundary line Re(s) = a.

In the region of convergence Re(s) > Re(s0), the Laplace transform of f can be expressed by integrating by parts as the integral

That is, F(s) can effectively be expressed, in the region of convergence, as the absolutely convergent Laplace transform of some other function. In particular, it is analytic.

There are several Paley–Wiener theorems concerning the relationship between the decay properties of f, and the properties of the Laplace transform within the region of convergence.

In engineering applications, a function corresponding to a linear time-invariant (LTI) system is stable if every bounded input produces a bounded output. This is equivalent to the absolute convergence of the Laplace transform of the impulse response function in the region Re(s) ≥ 0. As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part.

This ROC is used in knowing about the causality and stability of a system.

Properties and theorems

The Laplace transform has a number of properties that make it useful for analyzing linear dynamical systems. The most significant advantage is that differentiation becomes multiplication, and integration becomes division, by s (reminiscent of the way logarithms change multiplication to addition of logarithms).

Because of this property, the Laplace variable s is also known as operator variable in the L domain: either derivative operator or (for s−1)integration operator. The transform turns integral equations and differential equations to polynomial equations, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts to the original domain.

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: [22]

Properties of the unilateral Laplace transform
Time domains domainComment
Linearity Can be proved using basic rules of integration.
Frequency-domain derivativeF is the first derivative of F with respect to s.
Frequency-domain general derivativeMore general form, nth derivative of F(s).
Derivative f is assumed to be a differentiable function, and its derivative is assumed to be of exponential type. This can then be obtained by integration by parts
Second derivativef is assumed twice differentiable and the second derivative to be of exponential type. Follows by applying the Differentiation property to f′(t).
General derivativef is assumed to be n-times differentiable, with nth derivative of exponential type. Follows by mathematical induction.
Frequency-domain integration This is deduced using the nature of frequency differentiation and conditional convergence.
Time-domain integration u(t) is the Heaviside step function and (u  f)(t) is the convolution of u(t) and f(t).
Frequency shifting
Time shifting, u(t) is the Heaviside step function
Time scaling
Multiplication The integration is done along the vertical line Re(σ) = c that lies entirely within the region of convergence of F. [23]
Convolution
Complex conjugation
Cross-correlation
Periodic function f(t) is a periodic function of period T so that f(t) = f(t + T), for all t ≥ 0. This is the result of the time shifting property and the geometric series.
, if all poles of are in the left half-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 F(s) has a pole in the right-hand plane or poles on the imaginary axis (e.g., if or ), then the behaviour of this formula is undefined.

Relation to power series

The Laplace transform can be viewed as a continuous analogue of a power series. [24] If a(n) is a discrete function of a positive integer n, then the power series associated to a(n) is the series

where x is a real variable (see Z transform). Replacing summation over n with integration over t, a continuous version of the power series becomes

where the discrete function a(n) is replaced by the continuous one f(t).

Changing the base of the power from x to e gives

For this to converge for, say, all bounded functions f, it is necessary to require that ln x < 0. Making the substitution s = ln x gives just the Laplace transform:

In other words, the Laplace transform is a continuous analog of a power series, in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by es.

Relation to moments

The quantities

are the moments of the function f. If the first n moments of f converge absolutely, then by repeated differentiation under the integral,

This is of special significance in probability theory, where the moments of a random variable X are given by the expectation values . Then, the relation holds

Computation 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:

yielding

and in the bilateral case,

The general result

where denotes the nth derivative of f, can then be established with an inductive argument.

Evaluating integrals over the positive real axis

A useful property of the Laplace transform is the following:

under suitable assumptions on the behaviour of in a right neighbourhood of and on the decay rate of in a left neighbourhood of . The above formula is a variation of integration by parts, with the operators and being replaced by and . Let us prove the equivalent formulation:

By plugging in the left-hand side turns into:

but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted right-hand side.

This method can be used to compute integrals that would otherwise be difficult to compute using elementary methods of real calculus. For example,

Relationship to other transforms

Laplace–Stieltjes transform

The (unilateral) Laplace–Stieltjes transform of a function g : RR is defined by the Lebesgue–Stieltjes integral

The function g is assumed to be of bounded variation. If g is the antiderivative of f:

then the Laplace–Stieltjes transform of g and the Laplace transform of f coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the Stieltjes measure associated to g. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its cumulative distribution function. [25]

Fourier transform

The Laplace transform is similar to the Fourier transform. While the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. The Laplace transform is usually restricted to transformation of functions of t with t ≥ 0. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Techniques of complex variables can also be used to directly study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory. The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with imaginary argument s = or s = 2πfi [26] when the condition explained below is fulfilled,

This definition of the Fourier transform requires a prefactor of 1/(2π) on the reverse Fourier transform. This relationship between the Laplace and Fourier transforms is often used to determine the frequency spectrum of a signal or dynamical system.

The above relation is valid as stated if and only if the region of convergence (ROC) of F(s) contains the imaginary axis, σ = 0.

For example, the function f(t) = cos(ω0t) has a Laplace transform F(s) = s/(s2 + ω02) whose ROC is Re(s) > 0. As s = is a pole of F(s), substituting s = in F(s) does not yield the Fourier transform of f(t)u(t), which is proportional to the Dirac delta-function δ(ωω0).

However, a relation of the form

holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a weak limit of measures (see vague topology). General conditions relating the limit of the Laplace transform of a function on the boundary to the Fourier transform take the form of Paley–Wiener theorems.

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

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

Z-transform

The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution of

where T = 1/fs is the sampling period (in units of time e.g., seconds) and fs is the sampling rate (in samples per second or hertz).

Let

be a sampling impulse train (also called a Dirac comb) and

be the sampled representation of the continuous-time x(t)

The Laplace transform of the sampled signal xq(t) is

This is the precise definition of the unilateral Z-transform of the discrete function x[n]

with the substitution of zesT.

Comparing the last two equations, we find the relationship between the unilateral 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.

Borel transform

The integral form of the Borel transform

is a special case of the Laplace transform for f an entire function of exponential type, meaning that

for some constants A and B. The generalized Borel transform allows a different weighting function to be used, rather than the exponential function, to transform functions not of exponential type. Nachbin's theorem gives necessary and sufficient conditions for the Borel transform to be well defined.

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. [27] [28] For definitions and explanations, see the Explanatory Notes at the end of the table.

Because the Laplace transform is a linear operator,

Using this linearity, and various trigonometric, hyperbolic, and complex number (etc.) properties and/or identities, some Laplace transforms can be obtained from others more quickly than by using the definition directly.

The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t).

The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0). A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the region of convergence for causal systems is not the same as that of anticausal systems.

FunctionTime domain
Laplace s-domain
Region of convergenceReference
unit impulseall sinspection
delayed impulsetime shift of
unit impulse
unit stepRe(s) > 0integrate unit impulse
delayed unit stepRe(s) > 0time shift of
unit step
ramp Re(s) > 0integrate unit
impulse twice
nth power
(for integer n)
Re(s) > 0
(n > −1)
Integrate unit
step n times
qth power
(for complex q)
Re(s) > 0
Re(q) > −1
[29] [30]
nth rootRe(s) > 0Set q = 1/n above.
nth power with frequency shiftRe(s) > −αIntegrate unit step,
apply frequency shift
delayed nth power
with frequency shift
Re(s) > −αIntegrate unit step,
apply frequency shift,
apply time shift
exponential decay Re(s) > −αFrequency shift of
unit step
two-sided exponential decay
(only for bilateral transform)
α < Re(s) < αFrequency shift of
unit step
exponential approachRe(s) > 0Unit step minus
exponential decay
sine Re(s) > 0 Bracewell 1978 , p. 227
cosine Re(s) > 0 Bracewell 1978 , p. 227
hyperbolic sine Re(s) > |α| Williams 1973 , p. 88
hyperbolic cosine Re(s) > |α| Williams 1973 , p. 88
exponentially decaying
sine wave
Re(s) > −α Bracewell 1978 , p. 227
exponentially decaying
cosine wave
Re(s) > −α Bracewell 1978 , p. 227
natural logarithm Re(s) > 0 Williams 1973 , p. 88
Bessel function
of the first kind,
of order n
Re(s) > 0
(n > −1)
Williams 1973 , p. 89
Error function Re(s) > 0 Williams 1973 , p. 89
Explanatory notes:

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.

Here is a summary of equivalents:

s-domain equivalent circuits S-Domain circuit equivalents.svg
s-domain equivalent circuits

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.

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

Examples and applications

The Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant 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. The Laplace transform is invertible on a large class of functions. 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. [31]

The Laplace transform can also be used to solve differential equations and is used extensively in mechanical engineering and electrical engineering. The Laplace transform reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra. The original differential equation can then be solved by applying the inverse Laplace transform. English electrical engineer Oliver Heaviside first proposed a similar scheme, although without using the Laplace transform; and the resulting operational calculus is credited as the Heaviside calculus.

Evaluating improper integrals

Let . Then (see the table above)

In the limit , one gets

provided that the interchange of limits can be justified. This is often possible as a consequence of the final value theorem. Even when the interchange cannot be justified the calculation can be suggestive. For example, with a ≠ 0 ≠ b, proceeding formally one has

The validity of this identity can be proved by other means. It is an example of a Frullani integral.

Another example is Dirichlet integral.

Complex impedance of a capacitor

In the theory of electrical circuits, the current flow in a capacitor is proportional to the capacitance and rate of change in the electrical potential (in SI units). Symbolically, this is expressed by the differential equation

where C is the capacitance (in farads) of the capacitor, i = i(t) is the electric current (in amperes) 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.

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 V0 at zero:

Using this definition and the previous equation, we find:

which is the correct expression for the complex impedance of a capacitor. In addition, the Laplace transform has large applications in control theory.

Partial fraction expansion

Consider a linear time-invariant system with transfer function

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

To evaluate this inverse transform, we begin by expanding H(s) using the method of partial fraction expansion,

The unknown constants P and R are the residues located at the corresponding poles of the transfer function. Each residue represents the relative contribution of that singularity to the transfer function's overall shape.

By the residue theorem, the inverse Laplace transform depends only upon the poles and their residues. To find the residue P, we multiply both sides of the equation by s + α to get

Then by letting s = −α, the contribution from R vanishes and all that is left is

Similarly, the residue R is given by

Note that

and so the substitution of R and P into the expanded expression for H(s) gives

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.

Convolution

The same result can be achieved using the convolution property as if the system is a series of filters with transfer functions of 1/(s + a) and 1/(s + b). That is, the inverse of

is

Phase delay

Time functionLaplace 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:

This is just the sine of the sum of the arguments, yielding:

We can apply similar logic to find that

Statistical mechanics

In statistical mechanics, the Laplace transform of the density of states defines the partition function. [32] That is, the canonical partition function is given by

and the inverse is given by

See also

Notes

  1. 1 2 "Laplace Transform: A First Introduction". Math Vault. Retrieved 2020-08-08.
  2. "Differential Equations - Laplace Transforms". tutorial.math.lamar.edu. Retrieved 2020-08-08.
  3. 1 2 Weisstein, Eric W. "Laplace Transform". mathworld.wolfram.com. Retrieved 2020-08-08.
  4. "Des Fonctions génératrices" [On generating functions], Théorie analytique des Probabilités [Analytical Probability Theory] (in French) (2nd ed.), Paris, 1814, chap.I sect.2-20
  5. Jaynes, E. T. (Edwin T.) (2003). Probability theory : the logic of science. Bretthorst, G. Larry. Cambridge, UK: Cambridge University Press. ISBN   0511065892. OCLC   57254076.
  6. Abel, Niels H. (1820), "Sur les fonctions génératrices et leurs déterminantes", Œuvres Complètes (in French), II (published 1839), pp. 77–88 1881 edition
  7. Lerch, Mathias (1903), "Sur un point de la théorie des fonctions génératrices d'Abel" [Proof of the inversion formula], Acta Mathematica (in French), 27: 339–351, doi: 10.1007/BF02421315
  8. Heaviside, Oliver (January 2008), "The solution of definite integrals by differential transformation", Electromagnetic Theory, III, London, section 526, ISBN   9781605206189
  9. Bromwich, Thomas J. (1916), "Normal coordinates in dynamical systems", Proceedings of the London Mathematical Society , 15: 401–448, doi:10.1112/plms/s2-15.1.401
  10. An influential book was: Gardner, Murray F.; Barnes, John L. (1942), Transients in Linear Systems studied by the Laplace Transform, New York: Wiley
  11. Doetsch, Gustav (1937), Theorie und Anwendung der Laplacesche Transformation[Theory and Application of the Laplace Transform] (in German), Berlin: Springer translation 1943
  12. Euler 1744, Euler 1753, Euler 1769
  13. Lagrange 1773
  14. Grattan-Guinness 1997 , p. 260
  15. Grattan-Guinness 1997 , p. 261
  16. Grattan-Guinness 1997 , pp. 261–262
  17. Grattan-Guinness 1997 , pp. 262266
  18. Feller 1971 , §XIII.1
  19. The cumulative distribution function is the integral of the probability density function.
  20. Widder 1941 , Chapter II, §1
  21. Widder 1941 , Chapter VI, §2
  22. Korn & Korn 1967 , pp. 226227
  23. Bracewell 2000 , Table 14.1, p. 385
  24. Mattuck, Arthur. "Where the Laplace Transform comes from".
  25. Feller 1971 , p. 432
  26. Takacs 1953 , p. 93
  27. Riley, K. F.; Hobson, M. P.; Bence, S. J. (2010), Mathematical methods for physics and engineering (3rd ed.), Cambridge University Press, p. 455, ISBN   978-0-521-86153-3
  28. Distefano, J. J.; Stubberud, A. R.; Williams, I. J. (1995), Feedback systems and control, Schaum's outlines (2nd ed.), McGraw-Hill, p. 78, ISBN   978-0-07-017052-0
  29. Lipschutz, S.; Spiegel, M. R.; Liu, J. (2009), Mathematical Handbook of Formulas and Tables, Schaum's Outline Series (3rd ed.), McGraw-Hill, p. 183, ISBN   978-0-07-154855-7 – provides the case for real q.
  30. http://mathworld.wolfram.com/LaplaceTransform.html – Wolfram Mathword provides case for complex q
  31. Korn & Korn 1967 , §8.1
  32. RK Pathria; Paul Beal (1996). Statistical mechanics (2nd ed.). Butterworth-Heinemann. p.  56.

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Path integral formulation

The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

Radon transform

In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes. It was later generalized to higher-dimensional Euclidean spaces, and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function . The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° to every frequency component of a function, the sign of the shift depending on the sign of the frequency. The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal u(t). The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.

Dirichlet integral Integral of sin(x)/x from 0 to infinity.

In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:

In system analysis, among other fields of study, a linear time-invariant system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined below. These properties apply to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = x(t) ∗ h(t) where h(t) is called the system's impulse response and ∗ represents convolution. What's more, there are systematic methods for solving any such system, whereas systems not meeting both properties are generally more difficult to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.

In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind Jν(kr). The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r axis. The necessary coefficient Fν of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval.

In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. If f(t) is a real- or complex-valued function of the real variable t defined for all real numbers, then the two-sided Laplace transform is defined by the integral

In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.

Weierstrass transform "Smoothing" integral transform

In mathematics, the Weierstrass transform of a function f : RR, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x.

The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear PDEs belonging to the so-called integrable systems. It is named after Greek mathematician Athanassios S. Fokas.

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