Laplace transform
Also unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of complex analysis, and especially contour integrals, can be used for calculations.[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.[7] Joseph-Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form[10] 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.[13] Bernhard Riemann used the Laplace transform in his 1859 paper On the Number of Primes Less Than a Given Magnitude, in which he also developed the inversion theorem.Thomas Joannes Stieltjes considered a generalization of the Laplace transform connected to his work on moments.Other contributors in this time period included Mathias Lerch,[15] Oliver Heaviside, and Thomas Bromwich.Also during the 30s, the Laplace transform was instrumental in G H Hardy and John Edensor Littlewood's study of tauberian theorems, and this application was later expounded on by Widder (1941), who developed other aspects of the theory such as a new method for inversion.Edward Charles Titchmarsh wrote the influential Introduction to the theory of the Fourier integral (1937).The current widespread use of the transform (mainly in engineering) came about during and soon after World War II,[17] replacing the earlier Heaviside operational calculus.An important special case is where μ is a probability measure, for example, the Dirac delta function.In these cases, the image of the Laplace transform lives in a space of analytic functions in the region of convergence.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:[20]The Laplace transform can be alternatively defined in a purely algebraic manner by applying a field of fractions construction to the convolution ring of functions on the positive half-line.[22] 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.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.As a result, LTI systems are stable, provided that the poles of the Laplace transform of the impulse response function have negative real part.but assuming Fubini's theorem holds, by reversing the order of integration we get the wanted right-hand side.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.As s = iω0 is a pole of F(s), substituting s = iω in F(s) does not yield the Fourier transform of f(t)u(t), which contains terms proportional to the Dirac delta functions δ(ω ± ω0).The unilateral or one-sided Z-transform is simply the Laplace transform of an ideally sampled signal with the substitution ofComparing the last two equations, we find the relationship between the unilateral Z-transform and the Laplace transform of the sampled signal,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.The same result can be achieved using the convolution property as if the system is a series of filters with transfer functions 1/(s + α) and 1/(s + β).The wide and general applicability of the Laplace transform and its inverse is illustrated by an application in astronomy which provides some information on the spatial distribution of matter of an astronomical source of radiofrequency thermal radiation too distant to resolve as more than a point, given its flux density spectrum, rather than relating the time domain with the spectrum (frequency domain).Assuming certain properties of the object, e.g. spherical shape and constant temperature, calculations based on carrying out an inverse Laplace transformation on the spectrum of the object can produce the only possible model of the distribution of matter in it (density as a function of distance from the center) consistent with the spectrum.[40] When independent information on the structure of an object is available, the inverse Laplace transform method has been found to be in good agreement.Then the limit as x goes to infinity of e−x A(x) is equal to c. This statement can be applied in particular to the logarithmic derivative of Riemann zeta function, and thus provides an extremely short way to prove the prime number theorem.
mathematicsPierre-Simon Laplaceintegral transformfunctionvariabletime domaincomplexfrequency domaindifferentiationintegrationmultiplicationdivisionlogarithmsscienceengineeringdifferential equationsdynamical systemsordinary differential equationsintegral equationsalgebraic polynomial equationsconvolutionintegralcomplex numberFourier transformMellin transformFormallyanalytic functionpower seriesmomentscomplex analysiscontour integralsmathematicianastronomerPierre-Simon, Marquis de Laplaceprobability theorygenerating functionsz-transformcontinuous variableNiels Henrik AbelLeonhard Eulergamma functionJoseph-Louis Lagrangeprobability density functionsdifference equationJoseph FourierFourier seriesdiffusion equationperiodicCauchyoperational calculusOliver HeavisideBernhard RiemannOn the Number of Primes Less Than a Given MagnitudeRiemann zeta functionmodular transformation lawJacobi theta functionPoisson summationHjalmar MellinKarl Weierstrassspecial functionsThomas Joannes Stieltjeswork on momentsMathias LerchThomas BromwichRaymond PaleyNorbert WienerG H HardyJohn Edensor Littlewoodtauberian theoremsEdward Charles TitchmarshWorld War IIGustav Doetschdamped cosinesexponentially growingreal numberslocally integrableexponential typeLebesgue integralconditionally convergentimproper integralweak senseBorel measureprobability measureDirac delta functionLaplace–Stieltjes transformTwo-sided Laplace transformHeaviside step functionInverse Laplace transformLebesgue measureone-to-one mappingL∞(0, ∞)tempered distributionsanalytic functionsresidue theoremPost's inversion formulaapplied probabilityexpected valuerandom variableexpectationconventionmoment generating functionfirst passage timesstochastic processesMarkov chainsrenewal theorycumulative distribution functionfield of fractionsspace of abstract operatorsconverges absolutelyextended real constantdominated convergence theoremFubini's theoremMorera's theoremintegrating by partsPaley–Wiener theoremslinear time-invariant (LTI) systemLinearityDerivativedifferentiable functionmathematical inductionCircular convolutionComplex conjugationPeriodic functiongeometric seriesPeriodic summationInitial value theorem