Laplace transforms of various functions

There are several theorems that are useful in obtaining Laplace transforms of various functions. The first is the shifting theorem-. 

These theorems can be used to construct the Laplace transforms of various functions, and to find inverse transforms without carrying out an integral in the complex plane. 

The advantage of defining a transfer function in terms of Laplace transforms of input and output is that the differential equations developed to describe the unsteady-state behaviour of the system are reduced to simple algebraic relationships (e.g. cf. equations 7.17 and 7.19). Such relationships are much easier to deal with, and normal algebraic laws can be used to relate the various transfer functions of each component in the control loop (see Section 7.9). Furthermore, the output (or response) of the system to a variety of inputs may be obtained without classical integration. 

The characteristic function for a distribution law of a random variable is the Laplace transform of the expression of the distribution law. For the analysis of the properties of the distribution of a random variable, the characteristic function is good for the rapid calculation of the centred or not, momentum of various orders. Here below, we have the definition of the characteristic function cp (s) and its particularization with the case under discussion ... 

Determination of both the transmittance of the investigated object and the Laplace transform of the input functiony(r) furnishes the output function x(s)= y(s)- H(s). With the inverse transformation, we obtain y(t). Output functions x(t) of proportional, integrating and inertial objects for various input functions are collected in Table 2.1. 

After assuming a flow model based both on the physical structure of the reactor and the characteristic features of the experimental RTD curve, mass balance equations are written for the dispersion of an inert tracer among the various zones involved in the model. These equations are linear differential equations (ODE or PDE) owing to the linear character of mixing processes. By solving the equations in the Laplace domain, the theoretical transfer function G (s,p ) is obtained, which is nothing but the Laplace transform of the - theoretical RTD E (t,pj ) where pj are the parameters of the model... 

Various sets of functions with their Laplace transforms have been compiled in tables f42, 74], which can also be used for inverse transformation -S 1 (F(s) = F(f), etc. 

The traditional way is to measure the impedance curve, Z(co), point-after-point, i.e., by measuring the response to each individual sinusoidal perturbation with a frequency, to. Recently, nonconventional approaches to measure the impedance function, Z(a>), have been developed based on the simultaneous imposition of a set of various sinusoidal harmonics, or noise, or a small-amplitude potential step etc, with subsequent Fourier- and Laplace transform data analysis. The self-consistency of the measured spectra is tested with the use of the Kramers-Kronig transformations [iii, iv] whose violation testifies in favor of a non-steady state character of the studied system (e.g., in corrosion). An alternative development is in the area of impedance spectroscopy for nonstationary systems in which the properties of the system change with time. 

This function is the product of the KWW and power-law dependencies. The relaxation law (25) in time domain and the HN law (21) in the frequency domain are rather generalized representations that lead to the known dielectric relaxation laws. The fact that these functions have power-law asymptotes has inspired numerous attempts to establish a relationship between their various parameters [40,41]. In this regard, the exact relationship between the parameters of (25) and the HN law (21) should be a consequence of the Laplace transform according to (14) [11,12]. However, there is currently no concrete proof that this is indeed so. Thus, the relationship between the parameters of equations (21) and (25) seems to be valid only asymptotically. 

We make now, similarly as is common with the different integral transforms, a correspondence table between the stochastic variable and the associated characteristic function. Note, there are several integral transforms. The most well-known integral transformation might be the Fourier transform. Further, we emphasize the Laplace transform, the Mellin transform, and the Hilbert transform. These transformations are useful for the solution of various differential equations, in communications technology, all ranges of the frequency analysis, also for optical problems and much other more. We designate the stochastic variable with X. The associated characteristic function should be... 

Our formalism has led to a diffusive transport in the bulk [Eqs. (5) and (6)] coupled to an adsorption mechanism at the interface [Eq. (7)]. Yet unlike previous models, all of the equations have been derived from a single functional, and hence, various assumptions employed by previous works can be examined. Treating Eqs. (5) and (6) using the Laplace transform with respect to time, we obtain a relation similar to the Ward and Tordai result [1]. 

The various terms in the equation are as follows i is the current, n is the number of electrons transferred, F is the Faraday, A is the electrode area, C is the concentration, D is the diffusion coefficient, and t is the time. Thus the technique may be used to estimate, among other things, the charge transport parameter. The reader interested in the solution of the Fick s law equation using the Laplace transformation should consult Ref. 6. A closely related technique is chronocoulometry, in which the excitation function is still the potential pulse, but instead of monitoring the current, the integrated charge is monitored as a function of time. This entails less error as a cumulative measurement is made. The equation for chronocoulometry is... 

The t5q)ical resolution of the lifetime spectrometers is 200-250 ps, FWHM. This value is of the order of magnitude of the positron lifetimes in solids. Therefore the determination of the lifetime components r, is a delicate operation. Various methods are available (1) least-squares fit of a model spectrum convoluted with the resolution function of the spectrometer (Kirkegaard and Eldrup 1974) (2) Laplace transform (Gregory and Zhu 1990) and (3) maximum entropy (Shukla et al. 1993,1997). The last two methods provide... 

These equations are perturbed, linearized, and Laplace-transformed from the time domain to the frequency domain to evaluate the transfer functions between various thermal-hydraulic parameters. The Laplace-transformed equations are solved simultaneously by means of a matrix equation. 

The linearized and Laplace-transformed equations of the models described above are used to evaluate the various system transfer functions as functions of the Laplace variables s = cr + jco, where a is the real part and co is the imaginary part of the complex variable s. a refers to the damping constant (or damped exponential frequency) and co refers to the resonant oscillation frequency of the system. 

The decay is generally not in the form of a simple exponential decay function, but usually deviates from it because of various complexities in the fluid, such as particle size distribution in solution and/or multimodes of molecular motions. Those non-single exponential decays can be expressed as a linear superposition of monoexponential decays, weighted with a distribution frmction G(r, ), the spectrum of decay rates, resulting in a Laplace transformation. 

A complete exposition of the mathematical structure of linear viscoelasticity has been given by Gross [7]. Here we will only summarise certain parts of his argument to illustrate the use of Laplace and Fourier transforms in establishing the formal connections between various viscoelastic functions ... 

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