Useful Power Series

Particular functional forms can be found by using power series 106... 

Using power series expansions of functions to probe limiting behaviour for increasingly large or small values of the independent variable. 

Assuming that Mq(0 is analytic, a can be evaluated using power series expansion. 

Equation (6.147) is also valid for this same reaction as part of the two-step complex reaction in the domain of large time values, near the equilibrium of the complex reaction, as a result of the principle of detailed balance. Intuitively, this same relationship has to be valid in the domain of small time values in which the influence of the second reaction is insignificant. This can be checked using power series solutions to the kinetic equations, by successive derivation and substitution of initial values. Retaining the first two nonzero terms in the power series, Constales et al. (2012) found that... 

Another very useful power series expansion is the binomial expansion ... 

Remark 6.2. The change of variable (6.27) is only allowed in situations where the RHS of (6.29) remains bounded as a approaches 0 [58]. Here, this change of variables is allowed when R is away from 0, since after expanding (6.26) using power series, we get/(a cos p, —a sin p, a) = of a, p, a) for some bounded function /forO a aQ and for sufficiently small a 1 such that N 0. FI... 

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... 

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... 

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

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 ) ... 

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. 

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... 

Unnecessary loading of the breakdown fuse through transient overvoltages can be avoided by connection to a r element which consists of a length choke and transverse capacitors. So-called iron core chokes are most conveniently used for series chokes, which are usual in power electronics. A damping element with a 61-fjF capacitor is advised at the input and output of the r element. 

Generating functions are used in calculating moments of distributions for power series expansions. In general, the nth moment of a distribution,/fxj is E x ") = lx" f x) dx, where the integration is over the domain of x. (If the distribution is discrete, integration is replaced by summation.)... 

The combinatorial interpretation (or the computations of Sec. 14 regardless of combinatorial considerations) implies the following useful result Substituting a power series with non-negative integer coefficients in the difference of the cycle indices of A, and we get a power series with non-negative integer coefficients. 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

An essential improvement of the entire approach has recently been given by Kinoshita (1957), who pointed out that, instead of using Eq. III. 113, one could just as well use a power series expansion in the three variables... 

The moments of this-distribution can now be obtained by making use of the known power series expansion for e ... 

Expanding the second exponential in the right-hand side of Eq. (3-266) in a power series and making use of Eq. (3-261), we obtain... 

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