Series functional

This is the harmonic series, which we already found to diverge.  

A functional series has terms that are constants times functions. If a single independent variable is called x. 

There is an important mathematical concept called uniform convergence. If a functional series converges in some interval, it is imiformly convergent in that interval if it converges with at least a certain fixed rate of convergence in the entire interval. We do not discuss the details of this concept. If a functional series is uniformly convergent in some interval, it has been shown to have some useful mathematical properties, which we discuss later. 

There are two problems to be faced in constmcting a series to represent a given function. The first is finding the values of the coefficients so that the function will be correctly represented. The second is finding the interval in which the series is convergent and in which it represents the given function. 

A common type of fimctional series is the power series, in which the basis functions are powers of x — h, where x is the independent variable and his a. constant (it can equal zero). 

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The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

Again, this solution converges very slowly for small extents of loss, i.e., for small values of Stt/d1. In this case, the solution expressed as an error function series should be used (Appendix 8B)... 

The coefficients, bnkn, of JQ in this Bessel function series can be determined [8] ... 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

The solutions of differential equations system (D.l) with the boundary value problem given by Eqs. (D.4)-(D.5) can be written as the following functional series of the variable % with their coefficients being dependent on the variable, vp (i = O, R) ... 

By applying Koutecky s dimensionless parameter method [1], we suppose that solutions are functional series of the dimensionless variable... 

To solve the problem corresponding to the second potential step, the Koutecky s dimensionless parameter method [3] has been applied by assuming that the solutions of the differential equation system (G.29) are functional series of the dimensionless variables Xi and p ... 

Kapuy E (1967) The Physical Society of Japan (eds). Applicability of almost strongly orthogonal geminals in many-electron wave function. Series in Selected Papers in Physics, Theory of Molecular Structure II... 

Tetronic [BASF]. TM for a nonionic tetra-functional series of polyether block polymers ranging in physical form from liquids through pastes to finable solids. They are polyoxyalkylene derivatives of ethylenediamine. Physical state varies with molecular weight and oxyethylene content, 100% active. 

Turbo Method imports retention data and uploads methods. Includes peak tracking algorithm and results reporting functions. Series 200 HPLC with PDA and 50,000... 

A mathematical series is a sum of terms. A series can have a finite number of terms or can have an infinite number of terms. If a series has an infinite number of terms, an important question is whether it approaches a finite limit as more and more terms of the series are included (in which case we say that it converges) or whether it becomes infinite in magnitude or oscillates endlessly (in which case we say that it diverges). A constant series has terms that are constants, so that it equals a constant if it converges. Afunctional series has terms that are functions of one or more independent variables, so that the series is a function of the same independent variables if it converges. Each term of a functional series contains a constant coefficient that multiplies a function from a set of basis functions. The process of constructing a functional series to represent a specific function is the process of determining the coefficients. We discuss two common types of functional series, power series and Fourier series. 

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. 

An infinite functional series represents a function if it converges. 

There are some important mathematical questions about Fourier series, including the convergence of a Fourier series and the completeness of the basis functions. A set of basis functions is said to be complete for representation of a set of functions if a series in these functions can accurately represent any function from the set. We do not discuss the mathematics, but state the facts that were proved by Fourier (1) any Fourier series in is uniformly convergent for all real values of v (2) the set of sine and cosine basis functions in Eq. (6.39) is a complete set for the representation of periodic functions of period 2L. In many cases of functional series, the completeness of the set of basis functions has not been proved, but most people assume completeness and proceed. 

Other Functional Series with Orthogonal Basis Sets... 

Carrying out mathematical operations such as integration or differentiation on a functional series with a finite number of terms is straightforward, since no questions of convergence arise. However, carrying out such operations on an infinite... 

In this chapter we introduced mathematical series and mathematical transforms. A finite series is a sum of a finite number of terms, and an infinite series is a sum of infinitely many terms. A constant series has terms that are constants, and a functional series has terms that are functions. The two important questions to ask about a constant series are whether the series converges and, if so, what value it converges to. We presented several tests that can be used to determine whether a series converges. Unfortunately, there appears to be no general method for finding the value to which a convergent series converges. 

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. 

The other functional series that we discussed was the Fourier series, in which the basis functions are sine and cosine functions. This type of series is best suited for representing periodic functions and represents the function as a sum of the sine and cosine functions with the appropriate coefficients. The method of determining the coefficients to represent any particular function was given. 

For first-order irreversible reactions and Danckwerts residence time distribution Huang and Kuo derived two solutions one for long exposure times that expresses the concentration gradients in trigonometric function series and the following solution for rather short exposure times, obtained by Laplace transforms ... 

The shaded terms cancel out. Multiplying by the factor (1 —1/2 ) has evidently removed all the terms divisible by 2 from the zeta-function series. The next step is to multiply by (1 — 1/3 ). This gives... 

Resummation of asymptotic series The and y-functions are calculated perturbatively as series in the couplings Aj. The order of the expansion corresponds to the number of loops in the diagrammatic Feynman representation of the vertex functions (69). However, due to the (asymptotic) divergence of the RG functions series [84], one cannot directly derive reliable physical information... 

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