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Integral transform useful in probability theory, physics, and engineering
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable
Laplace_transform
following is a list of Laplace transforms for many common functions of a single variable. The Laplace transform is an integral transform that takes a function
List_of_Laplace_transforms
Mathematical operation
In mathematics, the inverse Laplace transform of a function F {\displaystyle F} is a real function f {\displaystyle f} that is piecewise-continuous,
Inverse_Laplace_transform
Mathematical operation
Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms
Two-sided_Laplace_transform
Linear transform from the time domain to the frequency domain
representation. It can be considered a discrete-time counterpart of the Laplace transform (the s-domain or s-plane). This similarity is explored in the theory
Z-transform
French polymath (1749–1827)
probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of
Pierre-Simon_Laplace
Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform.
Laplace–Stieltjes_transform
Mathematical operation
Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is
Mellin_transform
Variant of the Laplace integral transform
the Laplace–Carson transform, named for Pierre Simon Laplace and John Renshaw Carson, is an integral transform closely related to the standard Laplace transform
Laplace–Carson_transform
Generalization of the hypergeometric function
integral transforms like the Hankel transform and the Laplace transform and their inverses result when suitable G-function pairs are employed as transform kernels
Meijer_G-function
Indicator function of positive numbers
distributions. The Laplace transform of the Heaviside step function is a meromorphic function. Using the unilateral Laplace transform we have: H ^ ( s )
Heaviside_step_function
Mathematical model which is both linear and time-invariant
system is the Laplace transform or Z-transform of the system's impulse response, respectively. As a result of the properties of these transforms, the output
Linear_time-invariant_system
Relation between frequency- and time-domain behavior at large time
f ( t ) {\displaystyle f(t)} in continuous time has (unilateral) Laplace transform F ( s ) {\displaystyle F(s)} , then a final value theorem establishes
Final_value_theorem
Measure defined on all open sets of a topological space
Borel measure on the real line is of this kind. One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral
Borel_measure
Mapping involving integration between function spaces
frequency domain. Employing the inverse transform, i.e., the inverse procedure of the original Laplace transform, one obtains a time-domain solution. In
Integral_transform
Integral transform generalizing both Laplace and Sumudu transforms
mathematics, the Shehu transform is an integral transform which generalizes both the Laplace transform and the Sumudu integral transform. It was introduced
Shehu_transform
Mathematical transform that expresses a function of time as a function of frequency
spaces Hankel transform – Mathematical operation Hartley transform – Integral transform closely related to the Fourier transform Laplace transform – Integral
Fourier_transform
Signal processing conducted on analog signals
like the Fourier transform. The major difference is that the Laplace transform has a region of convergence for which the transform is valid. This implies
Analog_signal_processing
Integral transform
} The Laplace transform is the fractional Laplace transform when θ = 90 ∘ . {\displaystyle \theta =90^{\circ }.} The inverse Laplace transform corresponds
Linear canonical transformation
Linear_canonical_transformation
Function acting on function spaces
}^{+\infty }{g(\omega )\ e^{i\ \omega \ t}\ \mathrm {d} \ \omega }} The Laplace transform is another integral operator and is involved in simplifying the process
Operator_(mathematics)
Probability distribution
_{p}\left({\frac {\theta _{p}}{\theta _{q}}}-1\right).\end{aligned}}} The Laplace transform of the gamma PDF, which is the moment-generating function of the gamma
Gamma_distribution
Piecewise function that clamps its input to be non-negative
delta (in this formula, its derivative appears). The single-sided Laplace transform of R(x) is given as follows, L { R ( x ) } ( s ) = ∫ 0 ∞ e − s x R
Ramp_function
Function specifying the behavior of a component in an electronic or control system
dividing the Laplace transform of the output, Y ( s ) = L { y ( t ) } {\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}} , by the Laplace transform of the
Transfer_function
Integral expressing the amount of overlap of one function as it is shifted over another
f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u {\displaystyle
Convolution
Integral of sin(x)/x from 0 to infinity
improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign, contour
Dirichlet_integral
Mathematical computing environment
viewpoint=[path=M]); Laplace transform f := (1+A*t+B*t^2)*exp(c*t); ( 1 + A t + B t 2 ) e c t {\displaystyle \left(1+A\,t+B\,t^{2}\right)e^{ct}} inttrans:-laplace(f, t
Maple_(software)
Poisson point process
by ξ {\displaystyle \xi } , then η {\displaystyle \eta } has the Laplace transform L η ( f ) = exp ( − ∫ 1 − exp ( − f ( x ) ) ξ ( d x ) ) {\displaystyle
Cox_process
Method for solving linear differential equations using the Laplace transform
mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be
Laplace transform applied to differential equations
Laplace_transform_applied_to_differential_equations
Property of many linear time-invariant (LTI) systems
filter is u(t). Apply z-transform and Laplace transform on these two inputs to obtain the converted output signal. Perform z-transform on step input Z [ u
Infinite_impulse_response
Branch of engineering and mathematics
functions to functions of frequency by a transform such as the Fourier transform, Laplace transform, or Z transform. The advantage of this technique is that
Control_theory
Method of evaluating certain integrals along paths in the complex plane
contour are determined by its values along the contour. The inverse Laplace transform is defined by a complex contour integral known as the Bromwich integral:
Contour_integration
Mathematical identity in queueing theory
relationship between the queue length and service time distribution Laplace transforms for an M/G/1 queue (where jobs arrive according to a Poisson process
Pollaczek–Khinchine_formula
Unification of discrete and continuous theories of calculus
Laplace transform can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can
Time-scale_calculus
Laplace transform Inverse Laplace transform Two-sided Laplace transform Inverse two-sided Laplace transform Laplace–Carson transform Laplace–Stieltjes
List_of_transforms
Type of signal filter
poles and zeros of the Laplace transform in the complex plane. (In discrete time, one can similarly consider the Z-transform of the impulse response
Low-pass_filter
Theorem in complex analysis
which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function. If φ ( s )
Mellin_inversion_theorem
Electrical resonant circuit
^{2}}{\mathrm {d} t^{2}}}I(t)+\omega _{0}^{2}I(t)=0.} The associated Laplace transform is s 2 + ω 0 2 = 0 , {\displaystyle s^{2}+\omega _{0}^{2}=0,} thus
LC_circuit
Mathematical analysis of frequency content of signals
differential equations can be solved by a direct use of the Laplace transform. The Laplace transform for an M-dimensional case is defined as F ( s 1 , s 2
Multidimensional_transform
transforms include: Two-sided Laplace transform Mellin transform, another closely related integral transform Laplace transform: the Fourier transform
List of Fourier-related transforms
List_of_Fourier-related_transforms
Output of a dynamic system when given a brief input
impulse responses. The transfer function is the Laplace transform of the impulse response. The Laplace transform of a system's output may be determined by the
Impulse_response
on K = n {\displaystyle K=n} , a mixed Binomial processe has the Laplace transform L ( f ) = ( ∫ exp ( − f ( x ) ) P ( d x ) ) n {\displaystyle {\mathcal
Mixed_binomial_process
Opposition that a system presents to an acoustic pressure
inverse of R). Acoustic impedance, denoted Z, is the Laplace transform, or the Fourier transform, or the analytic representation of time domain acoustic
Acoustic_impedance
Generalization in fractional calculus
_{x}^{\alpha }}} is the Riemann–Liouville fractional derivative. The Laplace transform of the Caputo-type fractional derivative is given by: L x { a C D
Caputo_fractional_derivative
Type of random variable ordering
{E} [u(B)]} . Laplace transform order compares both size and variability of two random variables. Similar to convex order, Laplace transform order is established
Stochastic_ordering
Integral transform
}^{\infty }|f(t)|e^{-\sigma |t|}\,dt} is finite. For f ∈ Xσ, the Laplace transform of Iα f takes the particularly simple form ( L I α f ) ( s ) = s −
Riemann–Liouville_integral
Stochastic way of assigning quantities across a space
to be finite. For a random measure ζ {\displaystyle \zeta } , the Laplace transform is defined as L ζ ( f ) = E [ exp ( − ∫ f ( x ) ζ ( d x ) ) ]
Random_measure
Concept in probability theory and statistics
exponential order, the Fourier transform of f {\displaystyle f} is a Wick rotation of its two-sided Laplace transform in the region of convergence. See
Moment_generating_function
German mathematician (1892–1977)
1977) was a German mathematician best known for his research on the Laplace transform. He was also a decorated veteran of the First World War, and a pacifist
Gustav_Doetsch
Branch of mathematical analysis
which has the advantage that it is zero when f(t) is constant and its Laplace Transform is expressed by means of the initial values of the function and its
Fractional_calculus
Opposition of a circuit to a current when a voltage is applied
Signals are expressed in terms of complex frequency by taking the Laplace transform of the time domain expression of the signal. The impedance of the
Electrical_impedance
Electric circuit composed of resistors and capacitors
knowledge of the Laplace transform. The most straightforward way to derive the time domain behaviour is to use the Laplace transforms of the expressions
RC_circuit
Theorem in harmonic analysis
Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. It is of importance
Riemann–Lebesgue_lemma
Complex number representing a particular sine wave
some mathematical details, the phasor transform can also be seen as a particular case of the Laplace transform (limited to a single frequency), which
Phasor
Differential operator in mathematics
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean
Laplace_operator
Partial differential equations
the Green's function (or fundamental solution) for the Laplacian (or Laplace operator) in three variables is used to describe the response of a particular
Green's function for the three-variable Laplace equation
Green's_function_for_the_three-variable_Laplace_equation
Branch of mathematics
Fourier-related transforms Laplace transform (LT) Two-sided Laplace transform Mellin transform Non-uniform discrete Fourier transform (NDFT) Quantum Fourier
Fourier_analysis
"Smoothing" integral transform
Weierstrass transform exploits its connection to the Laplace transform mentioned above, and the well-known inversion formula for the Laplace transform. The result
Weierstrass_transform
Type of queue model in queueing theory
function of the first kind, obtained by using Laplace transforms and inverting the solution. The Laplace transform of the M/M/1 busy period is given by E (
M/M/1_queue
) . {\displaystyle \xi (A)\sim \operatorname {Bin} (n,P(A)).} The Laplace transform of a binomial process is given by L P , n ( f ) = [ ∫ exp ( − f
Binomial_process
Device for suppressing part of a signal
operated by the Laplace transform and its inverse (therefore, here below, the term "input signal" shall be understood as "the Laplace transform of" the time
Filter_(signal_processing)
British mathematician and electrical engineer (1850–1925)
new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, and rewrote Maxwell's equations
Oliver_Heaviside
Local pressure deviation caused by a sound wave
{p}}(s)} is the Laplace transform of sound pressure,[citation needed] Q ^ ( s ) {\displaystyle {\hat {Q}}(s)} is the Laplace transform of sound volume
Sound_pressure
Resistive and inductive circuit
_{R}}\end{aligned}}} The impulse response for each voltage is the inverse Laplace transform of the corresponding transfer function. It represents the response
RL_circuit
Curve for which the time to roll to the end is equal for all starting points
compute its Laplace transform, calculate the Laplace transform of d ℓ / d y {\displaystyle {d\ell }/{dy}} and then take the inverse transform (or try to)
Tautochrone_curve
Statistical physics approach
Kaniadakis Laplace transform (or κ-Laplace transform) is a κ-deformed integral transform of the ordinary Laplace transform. The κ-Laplace transform converts
Kaniadakis_statistics
Physical characteristic of oscillating systems
Vin(s) are the Laplace transform of the current and input voltage, respectively, and s is a complex frequency parameter in the Laplace domain. Rearranging
Resonance
Probability distribution
theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called
Laplace_distribution
Equation involving both integrals and derivatives of a function
\qquad x<0\end{array}}\right.} is the Heaviside step function. The Laplace transform is defined by, U ( s ) = L { u ( x ) } = ∫ 0 ∞ e − s x u ( x ) d x
Integro-differential_equation
Method for approximate evaluation of integrals
In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form ∫ a b e M f ( x ) d x , {\displaystyle
Laplace's_method
Method for converting signals between digital and analog
Laplace transform of integration of a function of time corresponds to simply multiplication by 1 s {\displaystyle {\tfrac {1}{\text{s}}}} in Laplace notation
Delta-sigma_modulation
Signal processing operation
that is an exact mapping of the z-plane to the s-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time
Bilinear_transform
Theorem in mathematics
holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform (see Mellin
Convolution_theorem
Resistor Inductor Capacitor Circuit
AC state behavior using the Laplace transform. If the voltage source above produces a waveform with Laplace-transformed V(s) (where s is the complex
RLC_circuit
Output as a function of input frequency
related to the transfer function in linear systems, which is the Laplace transform of the impulse response. They are equivalent when the real part σ
Frequency_response
Signal processing algorithm
Laplace transform) S x + ( s ) {\displaystyle S_{x}^{+}(s)} is the causal component of S x ( s ) {\displaystyle S_{x}(s)} (i.e., the inverse Laplace transform
Wiener_filter
Diagram showing the singularities of a given control system's transfer function
continuous-time or a discrete-time system: Continuous-time systems use the Laplace transform and are plotted in the s-plane: s = σ + j ω {\displaystyle s=\sigma
Pole–zero_plot
Mathematical theorem
^{n}} . This extension of the Fourier transform to the complex domain is called the Fourier–Laplace transform. Schwartz's theorem—An entire function
Paley–Wiener_theorem
Tool for studying defects in semiconductors
There is an extension to DLTS known as a high resolution Laplace transform DLTS (LDLTS). Laplace DLTS is an isothermal technique in which the capacitance
Deep-level transient spectroscopy
Deep-level_transient_spectroscopy
Technique in mathematics
eigenspace of A. The Hille–Yosida theorem relates the resolvent through a Laplace transform to an integral over the one-parameter group of transformations generated
Resolvent_formalism
Time-frequency transform in geophysics
transform requires specific tools like standard multiresolution analysis. Geophysical signal analysis Reflection seismology Global seismology Laplace
S_transform
Formal and systematic written discourse on some subject
century and was ultimately developed into the modern theory of the Laplace transform, now of ubiquitous usage and a standard part of the undergraduate
Treatise
applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk
Starred_transform
Mathematical function common in physics
Wuttke, J. (2012). "Laplace–Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source
Stretched exponential function
Stretched_exponential_function
Second-order partial differential equation
mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties
Laplace's_equation
Operator generalizing the Laplacian in differential geometry
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space
Laplace–Beltrami_operator
Concept in mathematics
integral transforms into more abstract spaces, vector-valued functions, and operator spaces. Examples of such extensions include vector-valued Laplace transforms
Bochner_integral
Mathematical theorem using Laplace transform
{\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt} be the (one-sided) Laplace transform of ƒ(t). If f {\displaystyle f} is bounded on ( 0 , ∞ ) {\displaystyle
Initial_value_theorem
Signal representation
systems. Fourier transform – aperiodic signals, transients. Laplace transform – electronic circuits and control systems. Z transform – discrete-time signals
Frequency_domain
Differentiation under the integral sign formula
of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be
Leibniz_integral_rule
Mathematical theorem
exponential functions or in more abstract language, that it is the Laplace transform of a positive Borel measure on [0, ∞). In one important special case
Bernstein's theorem on monotone functions
Bernstein's_theorem_on_monotone_functions
Type of differential equation
Robin boundary condition Cauchy problem Various topics Jet bundle Laplace transform applied to differential equations List of dynamical systems and differential
Partial_differential_equation
Mathematical signal manipulation by computers
oscillate. The Z-transform provides a tool for analyzing stability issues of digital IIR filters. It is analogous to the Laplace transform, which is used
Digital_signal_processing
Mathematical operation
the Hankel transform and its inverse work for all functions in L2(0, ∞). The Hankel transform can be used to transform and solve Laplace's equation expressed
Hankel_transform
Determining all voltages and currents within an electrical network
is usual practice to carry out a Laplace transform on them first and then express the result in terms of the Laplace parameter s, which in general is
Network analysis (electrical circuits)
Network_analysis_(electrical_circuits)
Fundamental principle of physics
mathematical techniques, frequency-domain linear transform methods such as Fourier and Laplace transforms, and linear operator theory, that are applicable
Superposition_principle
Mathematical model of a system in control engineering
matrix form, offering a compact alternative to the frequency domain’s Laplace transforms for multiple-input and multiple-output (MIMO) systems. Unlike the
State-space_representation
Mathematical method in calculus
used to find the Laplace transform of a derivative of a function. The above result tells us about the decay of the Fourier transform, since it follows
Integration_by_parts
Sub-discipline of electrical engineering
properties of Laplace transform, continuous-time and discrete-time Fourier series, continuous-time and discrete-time Fourier Transform, z-transform. Sampling
Electronics_engineering
Flow graph invented by Claude Shannon
of some parameter like the Laplace transform variable s. Signal-flow graphs are very often used with Laplace-transformed signals, because then they represent
Signal-flow_graph
LAPLACE TRANSFORM
LAPLACE TRANSFORM
Female
Greek
(Λαλαγη) Classical Greek name derived from the word lalagein, LALAGE means "to babble."Â
Boy/Male
Sikh
Palace
Girl/Female
Indian, Sanskrit
Place
Boy/Male
British, English, French, German
Place
Girl/Female
Indian, Punjabi, Sikh
Palace
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Tamil, Telugu
Palace
Boy/Male
Hindu, Indian, Tamil
Place
Male
English
English surname transferred to forename use, from an ethnic byname, from Old French waleis, WALLACE means "foreigner, stranger," especially Celtic or Roman.
Girl/Female
Indian
Place
Girl/Female
Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
Palace
Girl/Female
Greek
Babble. Verbose.
Boy/Male
American, Anglo, Australian, British, Chinese, Christian, English, French, German, Indian, Scottish, Teutonic
Welshman; Stranger; Foreign; Celtic; From Wales
Girl/Female
Australian, Christian, Greek
Blabber; Prattler
Girl/Female
Tamil
Palace
Girl/Female
Tamil
Kshetra | கà¯à®·à¯‡à®¤à¯à®°Â
Place
Kshetra | கà¯à®·à¯‡à®¤à¯à®°Â
Boy/Male
Anglo Saxon American English Teutonic German Scottish
Stranger.
Boy/Male
Greek, Hindu, Indian, Russian
Place
Boy/Male
Christian & English(British/American/Australian)
Stranger
Girl/Female
Hindu, Indian
Place
Boy/Male
English
Place.
LAPLACE TRANSFORM
LAPLACE TRANSFORM
Boy/Male
Sikh
Friend of the Lord Sun
Boy/Male
Hindu, Indian
Fearless
Girl/Female
Greek
Wise.
Girl/Female
English
Warm.
Girl/Female
American, Australian, British, English, Greek, Norse
From the Hall; Army Power
Girl/Female
American, Australian, British, English, French, Welsh
Protector of the Sea; Sea Lord; Great Ruler; Guardian from the Sea
Boy/Male
Tamil
Soumyajit | ஸோஉஂமà¯à®¯à®¾à®œà¯€à®¤
One who won beauty
Girl/Female
Hebrew
Bird.
Female
Native American
 Native American Algonquin name KIMI means "secret." Compare with another form of Kimi.
Boy/Male
Assamese, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Traditional
Sweet Like Moon
LAPLACE TRANSFORM
LAPLACE TRANSFORM
LAPLACE TRANSFORM
LAPLACE TRANSFORM
LAPLACE TRANSFORM
n.
To put or set in a particular rank, office, or position; to surround with particular circumstances or relations in life; to appoint to certain station or condition of life; as, in whatever sphere one is placed.
n.
To put out at interest; to invest; to loan; as, to place money in a bank.
v. t.
To refund; to repay; to restore; as, to replace a sum of money borrowed.
n.
To set; to fix; to repose; as, to place confidence in a friend.
n.
To assign a place to; to put in a particular spot or place, or in a certain relative position; to direct to a particular place; to fix; to settle; to locate; as, to place a book on a shelf; to place balls in tennis.
n.
A retired or private place.
n.
The residence of a sovereign, including the lodgings of high officers of state, and rooms for business, as well as halls for ceremony and reception.
n.
A broad dagger formerly worn at the girdle.
n.
Loosely, any unusually magnificent or stately house.
v. t.
To take the place of; to supply the want of; to fulfull the end or office of.
n.
The official residence of a bishop or other distinguished personage.
v. t.
To place again; to restore to a former place, position, condition, or the like.
v. t.
To supply or substitute an equivalent for; as, to replace a lost document.
n.
Ordinal relation; position in the order of proceeding; as, he said in the first place.
n.
Reception; effect; -- implying the making room for.
n.
To attribute; to ascribe; to set down.
n.
Position in the heavens, as of a heavenly body; -- usually defined by its right ascension and declination, or by its latitude and longitude.
n.
See Haut pas.
v. t.
To put in a new or different place.