Series power

Almost all functions can be represented by power series of the form 

Useful results can be obtained when power series are differentiated or integrated term by term. This is a valid procedure under very general conditions. 

With arctan 1 = r/4, this gives a famous series for tt 

A second example begins with another binomial expansion 

This give a series representation for the natural logarithm  

It is often helpful to express a function in terms of a power series. A very simple power series is 

The coefficients, a, are often related to one another in a simple way which is determined by the nature of the function. An important method of expressing functions in a power series is the Taylor and Maclaurin expansions. In a Taylor expansion the function f(x) is expanded about a given point xq and the coefficients are related to the values of the derivatives of the function at x = xq. Thus, the Taylor expansion of f(x) is 

The Maclaurin expansion is the special case in which the function is expanded about X = 0. Thus, the Maclaurin expansion of/(x) is 

A list of important functions expressed as power series follows  

These expansions are especially useful when ax I. Under these circumstances the function can be approximated by the first few terms in its expansion. 

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The virial equation of state is a power series in the reciprocal molar volume or in the pressure ... 

Figure 1. Phase delay of Licrilite E202 cell, thickness 9.47)im, X 5l4nm. The solid line shows a power series best fit for the cell average data.

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

Theory shows that these equations must be simple power series in the concentration (or an alternative composition variable) and experimental data can always be fitted this way.)... 

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

Figure A2.5.16 shows the ooexistenoe ourve obtained from equation (A2.5.16). The logaritluns (or the hyperbolio tangent) oan be expanded in a power series, yielding...

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

The exponential fiinction of the matrix can be evaluated tln-ough the power series expansion of exp(). c is the coliinm vector whose elements are the concentrations c.. The matrix elements of the rate coefficient matrix K are the first-order rate constants W.. The system is called closed if all reactions and back reactions are included. Then K is of rank N- 1 with positive eigenvalues, of which exactly one is zero. It corresponds to the equilibrium state, witii concentrations r detennined by the principle of microscopic reversibility ... 

The electronic transition moment of equation (B1.1.5) is related to the intensity that the transition would have if the nuclei were fixed in configuration Q, but its value may vary with that configuration. It is often usefiil to expand Pi CQ) as a power series in the nonnal coordinates, Q. ... 

We then write the solution of equation B1.5.7 as a power series expansion hi temis of the strength X of the perturbation ... 

If the polarization of a given point in space and time (r, t) depends only on the driving electric field at the same coordmates, we may write tire polarization as P = P(E). In this case, we may develop the polarization m power series as P = = P - + P - + P - +, where the linear temi is = X] Jf/ Pyand the... 

Although all of the nuclear coordinates participate in this kinetic energy operator, and in our previous discussions, all of the nuclear coordinates are expanded, with respect to an equivalent position, in power series of the parameter K, here in the specific case of a diatomic molecule, we found that only the R coordinate seems to have an equilibrium position in the molecular fixed coordinates. This means that actually we only have to, or we can only, expand the R coordinate, but not the other coordinates, in the way that... 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

Assuming now that the power series expansion in F(p) can be terminated to keep /f(p) well behaved at large p values, it may be shown [95] that... 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

Having found solutions at these limits, we will use a power series in p to "interpolate" between these two limits. 

Expressing p(R) in a power series expansion about the equilibrium bond length position (denoted Rg collectively and Ra,g individually) ... 

Beyond sueh eleetronie symmetry analysis, it is also possible to derive vibrational and rotational seleetion rules for eleetronie transitions that are El allowed. As was done in the vibrational speetroseopy ease, it is eonventional to expand if (R) in a power series about the equilibrium geometry of the initial eleetronie state (sinee this geometry is more eharaeteristie of the moleeular strueture prior to photon absorption) ... 

The desired physieal property must be extraeted from the power series expansion of AE in powers of V. 

Equations for the order-by-order eorreetions to the wavefunetions and energies are obtained by using these power series expressions in the full Sehrodinger equation ... 

Using the faet that k is an eigenfunetion of HD and employing the power series expansion of /k allows one to generate the fundamental relationships among the energies Ek ) and the wavefunetions /kl ) ... 

To extraet from this set of coupled equations relations that ean be solved for the eoeffieients , whieh embodies the desired wavefunetion perturbations /k ", one eolleets together all terms with like power of X in the above general equation (in doing so, it is important to keep in mind that Ek itself is given as a power series in X). 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

By describing the concentration dependence of an observable property as a power series, Eq. (9.9) plays a comparable role for viscosity as Eq. (8.83) does for osmotic pressure. 

Equation (10.82) is a correct but unwieldy form of the Debye scattering theory. The result benefits considerably from some additional manipulation which converts it into a useful form. Toward this end we assume that the quantity srj, is not too large, in which case sin (srj, ) can be expanded as a power series. Retaining only the first two terms of the series, we obtain... 

However, unlike electrical anharmonicity, mechanical anharmonicity modifies the vibrational term values and wave functions. The harmonic oscillator term values of Equation (6.3) are modified to a power series in (u + ) ... 

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