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The Laplace transform is often used in circuit analysis, and simple conversions to the sdomain of circuit elements can be made. Circuit elements can be transformed into impedances, very similar to phasor impedances.
Here is a summary of equivalents:
Note that the resistor is exactly the same in the time domain and the sdomain. 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 sdomain account for that.
The equivalents for current and voltage sources are simply derived from the transformations in the table above.
The Laplace transform is used frequently in engineering and physics; the output of a linear timeinvariant 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.
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.
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 V_{0} 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.
Consider a linear timeinvariant 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.
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
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:
This is just the sine of the sum of the arguments, yielding:
We can apply similar logic to find that
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
In engineering, a transfer function of an electronic or control system component is a mathematical function which theoretically models the device's output for each possible input. In its simplest form, this function is a twodimensional graph of an independent scalar input versus the dependent scalar output, called a transfer curve or characteristic curve. Transfer functions for components are used to design and analyze systems assembled from components, particularly using the block diagram technique, in electronics and control theory.
In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.
In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle of the summation can be made to approximate an arbitrary function in that interval. As such, the summation is a synthesis of another function. The discretetime Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.
In mathematics and signal processing, the Ztransform converts a discretetime signal, which is a sequence of real or complex numbers, into a complex frequencydomain representation.
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
In fractional calculus, an area of mathematical analysis, the differintegral is a combined differentiation/integration operator. Applied to a function ƒ, the qdifferintegral of f, here denoted by
The Laplace–Stieltjes transform, named for PierreSimon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For realvalued functions, it is the Laplace transform of a Stieltjes measure, however it is often defined for functions with values in a Banach space. It is useful in a number of areas of mathematics, including functional analysis, and certain areas of theoretical and applied probability.
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the twosided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.
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 quantummechanically possible trajectories to compute a quantum amplitude.
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 (twodimensional) 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 higherdimensional 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 crosssectional 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 realvalued 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.
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 timeinvariant system is a system that produces an output signal from any input signal subject to the constraints of linearity and timeinvariance; 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 J_{ν}(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 F_{ν} 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 twosided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Twosided Laplace transforms are closely related to the Fourier transform, the Mellin transform, and the ordinary or onesided Laplace transform. If f(t) is a real or complexvalued function of the real variable t defined for all real numbers, then the twosided 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 twodimensional 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.
In mathematics, the Weierstrass transform of a function f : R → R, 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 socalled integrable systems. It is named after Greek mathematician Athanassios S. Fokas.
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