Coefficients of the power series

These statements are a consequence of the recursion relations obtained by identifying the coefficients of the power series expansion on the right- and left-hand side of the equation. For example, in (4.6), the coefficient of x" is (n > 1) on the left-hand side, and on the right-hand side a polynomial in R, . [cf. (2.56)], which implies the uniqueness. The coefficients of the polynomial mentioned are non-negative the term occurs, coming from x/, thus Rj > n-i statements that the coefficients are... 

Since the coefficients of the power series r(x) are all positive, the point X = p must be singular thus, equation (4.25) holds for x = p. The coefficients of the power series on the left-hand side of (4.25) are all positive, except for the constant, which is equal to 0. On the circle of convergence x = p, the absolute value of the series assumes, therefore, its maximum at x = p. We conclude that x p is the only solution of (4.25) on the circle of convergence the point... 

While I am no longer working in this field, and cannot easily do simulations, I think that a 2 factor PCR or PLS model would fully model the simulated spectra. At any wavelength in your simulation, a second degree power series applies, which is linear in coefficients, and the coefficients of a 2 factor PCR or PLS model will be a linear function of the coefficients of the power series. (This assumes an adequate number of calibration spectra, that is, at least as many spectra as factors and a sufficient number of wavelength, which the full spectrum method assures.) The PCR or PLS regression should find the linear combination of these PCR/PLS coefficients that is linear in concentration. 

The coefficients of the power series expansion of the extinction efficiency, to a 5th order approximation, were derived by Penndorf ( 17) and independently at our laboratories using Macsyma (18). The coefficients of the series are ... 

One way to handle a curved calibration line is to fit the line to a power series. A cubic equation (y = a + bx + cx + dx ) is usually sufficient to fit a case such as Figure 21-1. (In any event, since there are only six known points, you couldn t use a polynomial with more than five adjustable parameters.) You can use either LI NEST or the Solver to obtain the coefficients of the power series. Figure 21-2 shows a spreadsheet in which LI NEST is used to find the regression coefficients for the equation Rdg = a + bx ppm + c x (ppm) + dx (ppm) ... 

In contrast, the force of interaction between two ions is long-range and at large distances is proportional to 1/r, where r is the distance between the ions. Thus the solution cannot be considered to be composed of noninteracting clusters, and power series expansions in concentration are not possible. Statistical mechanical treatments of this problem demonstrate that the coefficients of the power series expansions diverge for coulomb forces and that another representation for the properties of the solution must be found. The rigorous molecular considerations confirm the results of the Debye-Huckel treatment for dilute solutions and demonstrate that the assumptions of the Arrhenius hypothesis are incorrect. 

Inserting expressions (36)-(39) into Eq. (32a) in which and are multiplied by s, and collating terms of equal powers of r, gives the following set of equations for the coefficients of the power series for the flux difference ... 

Table 11.1 Coefficients of the power series expansion of the unitary transformation U for five different parametrizations [606,611]. The first two coefficients have been fixed to be o = = 1- Note that all coefficients are only given with an accuracy of three digits after the...

A new look at the thermodynamics of structural phase transitions has been provided by the Landau theory (Landau and Lifshitz, 1980 Toledano and Toledano, 1987). Its principal element is a Gibbs free energy expansion in the form of a Taylor series in an order parameter Q. As the value of Q cannot be directly assessed, the coefficients of the power series... 

The recurrent formula, linking the coefficients of the power series (4.3) with the virial coefficients of Eq. (0.7), is presented in Ref. [0.5]. In this case... 

Asymptotic Values of the Coefficients of Certain Power Series... 

In the preceding section the analytical behavior of the power series q x) r(x), s(x), t(x) on the circle of convergence has been examined. An easily derived and well-known relation between the singularities and the coefficients of a power series is given below it allows for several inferences. ... 

In the preceding paragraphs of this section we have summed the terms arising from the partial expansion of the exponentials occurring in the coefficients of the powers of particle concentrations to obtain a series of multiple infinite sums, the terms of which are convergent. The terms in S(R) are of the same form as those in the Mayer solution theory, apart from replacement of integration by summation and the fact that mu differs from the solution value because of the discreteness of the lattice. The evaluation of wi - is outlined in the next section. It is found that the asymptotic form is... 

Here we give explicit expressions for the coefficients in the power series expansions of the potential, the azimuthal angle, and the vector potential. The coefficients in the quadratic expansion, (3.13), of the potential are... 

If this series is properly convergent, we know that in order for it to equal zero for all values of X, the coefficients of the powers of X must vanish separately.1 fDhe coefficient of X° when equated to zero gives Equation 23-4, ho that we were justified in beginning the expansions 23-6 and 23 7 with the terms [Pg.158]

The radius of convergence Rc of the power series depends strongly on the choice of the odd coefficients as may be demonstrated by the following three examples, which can be given in closed form ... 

An integral transform is similar to a functional series, except that it contains an integration instead of a summation, which corresponds to an integration variable instead of a summation index. The integrand contains two factors, as does a term of a functional series. The first factor is the transform, which plays the same role as the coefficients of a power series. The second factor is the basis function, which plays the same role as the set of basis functions in a functional series. We discuss two types of transforms, Fourier transforms and Laplace transforms. 

We now use a fact about series [Eq. (9) of Appendix C] If two power series in the same independent variable are equal to each other for all values of the independent variable, then any coefficient in one series is equal to the corresponding coefficient of the other series. 

A functional series is one way of representing a function. Such a series consists of terms, each one of which is a basis function times a coefficient. A power series uses powers of the independent variable as basis functions and represents a function as a sum of the appropriate linear function, quadratic function, cubic function, etc. We discussed Taylor series, which contain powers of x — h, where h is a constant, and also Maclaurin series, which are Taylor series with h =0. Taylor series can represent a function of x only in a region of convergence centered on h and reaching no further than the closest point at which the function is not analytic. We found the general formula for determining the coefficients of a power series. 

Since in the perturbed motion Jf° can deviate but slightly from the corresponding action variable J/=0, we can consider and ij to be small. If we substitute the new variables in the Hamiltonian function, we can expand this in terms of f and rj in such a way that each coefficient of the powers of A will itself be a series in increasing powers of f and rj°. 

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