Series representation

The volumetric properties of fluids are conveniently represented by PVT equations of state. The most popular are virial, cubic, and extended virial equations. Virial equations are infinite series representations of the compressibiHty factor Z, defined as Z = PV/RT having either molar density, p[ = V ), or pressure, P, as the independent variable of expansion ... 

Virial Equations of State The virial equation in density is an infinite-series representation of the compressiDility factor Z in powers of molar density p (or reciprocal molar volume V" ) about the real-gas state at zero density (zero pressure) ... 

The simplest form of approximation to a continuous function is some polynomial. Continuous functions may be approximated in order to provide a simpler form than the original function. Truncated power series representations (such as the Taylor series) are one class of polynomial approximations. 

Predescu, C. Doll, J. D., Optimal series representations for numerical path integral simulations, J. Chem. Phys. 2003,117, 7448-7463... 

The variable x is usually the mole fraction of the components. The last expression was first introduced by Guggenheim [5]. Equation (3.60) is a particular case of the considerably more general Taylor series representation of Y as shown by Lupis [6]. Let us apply a Taylor series to the activity coefficient of a solute in a dilute binary solution ... 

There is an alternative way to generate the Nyquist plots that is often more convenient to use, particularly in high-order systems. Equation (18.13) gives a doubly infinite series representation of. ... 

The coefficients of the sines and cosines will be real for real data. Restoring a high-frequency band of c (unique complex) discrete spectral components to a low-frequency band of b (unique complex) spectral components will be the same (when transformed) as forming the discrete Fourier series from the high-frequency band and adding this function to the series formed from the low-frequency band. When applying the constraints in the spatial domain, the Fourier series representation will be used. 

The sum u(k) + v(k) yields the restored function. The coefficients in v(k) are varied to satisfy the constraints. Because u(k) is constant for all function continuations discussed here, no useful purpose is served by writing out its Fourier series expression, and so the series representation will always be suppressed. 

However, there are important advantages to iterative methods when the number of equations to be solved is large. Once the coefficients have been obtained, they may be converted to complex form and added to the original spectrum. Taking the inverse DFT would then yield the restored function. However, if the number of solved coefficients is small, it may be quicker simply to substitute the coefficients into the series representation for v(k) and add this series to u(k). 

Figure 7.81 compares the outputy of the on-off element over the period 0 to 2n with the values of the first two terms of the equivalent Fourier series representation (equation 7.196). Clearly the most significant contribution is given by the fundamental component S) sin cot. Moreover, the higher frequency contributions of Sj sin cot, 5 sin cot etc. are progressively attenuated more by other linear components in the system and thus have less effect on the operating characteristics of the system. Hence, the output of the non-linearity is well represented by the fundamental component, i.e. ... 

Let us briefly examine the importance of the Mittag-Leffier function in relaxation modelling. The mathematical properties of the Mittag-Leffier function are compiled in Appendix B. Besides via the series representation, the Mittag-Leffier function is defined through its Laplace transform... 

Since all renormalized quantities exist for d < 4 this implies that the ratios zPfzP etc have a finite limit for d = 4. Using the power series representation of the Z-factors we can in principle solve Eq. (11.87 i) to find as power series in Substituting the result into Eqs, (11-87 ii), (11.87 iii)... 

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

An alternative way of computing the coefficients [p , -r and q , -r comes from observing that the two power series representations in Eq. (114) are the truncated Maclaurin expansions. Thus,... 

The orthogonal characteristic polynomials or eigenpolynomials Qn(u) play one of the central roles in spectral analysis since they form a basis due to the completeness relation (163). They can be computed either via the Lanczos recursion (84) or from the power series representation (114). The latter method generates the expansion coefficients q , -r through the recursion (117). Alternatively, these coefficients can be deduced from the Lanczos recursion (97) for the rth derivative Q /r(0) since we have qni r = (l/r )Q r(0) as in Eq. (122). The polynomial set Qn(u) is the basis comprised of scalar functions in the Lanczos vector space C from Eq. (135). In Eq. (135), the definition (142) of the inner product implies that the polynomials Qn(u) and Qm(u) are orthogonal to each other (for n= m) with respect to the complex weight function dk, as per (166). The completeness (163) of the set Q (u) enables expansion of every function f(u) e C in a series in terms of the... 

The polynomials P F(u) and Q7(u) from Eq. (240) are given by their general power series representations as ... 

W.M. Huo, Convergent series representation for the generalized oscillator strength of electron-impact ionization and an improved binary-encounter-dipole model, Phys. Rev. A 64 (2001) 042719. 

For many purposes it is desirable to constract an arbitrary series representation of the function y = Ax) over the available range of x. There are several ways of doing this, of which two are very common. In both methods smoothing is often important as a way to avoid erratic fluctuations that may necessitate artificially high-order series. 

Hint the coth(a) function can be expressed in a series representation as ... 

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