Taylors theorem

So the FC integral is added to the very few physical systems [18] which are realizations of this particular algebra. Using the Taylor theorem for shift operators due to Sack [19], and the Cauchy relation mentioned above, we can apply this very general idea to the specific case of the harmonic oscillator to obtain the closed formula (5). Recurrence relations can also be obtained by noticing that O is in reality a superoperator which maps normal ladder operators by the canonical transformation ... 

From the Taylor theorem, we can derive the difference approximation for the first- and second-order derivatives by expanding y about x,.When using central differences, the first-order derivative is approximated by... 

TayW43 Taylor, W. J. Applications of Polya s Theorem to optical, geometrical and structural isomerism. J. Chem. Phys. 11 (1943) 532. 

This argument tacitly assumes that J"M h t) kdt < oo for all positive mtegers A more careful argument, based on Taylor s theorem with the remainder, shows that it is sufficient that this condition be fulfilled for k a= 1,2. 

The conditions which must be satisfied at the plait point may be deduced as follows Expand by Taylor s theorem the expressions on the right of (9) and (10), omitting terms of higher orders than the second ... 

This is the BINOMIAL THEOREM. Using a Taylor Expansion, we can find the total probability for items taken r at a time as ... 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

From (9.27), we see that this approach will work nicely if the variance is always small Taylor s theorem with remainder tells us that the error of the first-derivative - mean-field - contribution is proportional to the second derivative evaluated at an intermediate A. That second derivative can be identified with the variance as in (9.27). If that variance is never large, then this approach should be particularly effective. For further discussion, see Chap. 4 on thermodynamic integration, and Chap. 6 on error analysis in free energy calculations. 

Similarly, many different types of functions can be used. Arden discusses, for example, the use of Chebyshev polynomials, which are based on trigonometric functions (sines and cosines). But these polynomials have a major limitation they require the data to be collected at uniform -intervals throughout the range of X, and real data will seldom meet that criterion. Therefore, since they are also by far the simplest to deal with, the most widely used approximating functions are simple polynomials they are also convenient in that they are the direct result of applying Taylor s theorem, since Taylor s theorem produces a description of a polynomial that estimates the function being reproduced. Also, as we shall see, they lead to a procedure that can be applied to data having any distribution of the X-values. 

We will apply these to a small control volume of radius b and height dz, as shown in Figure 10.5, and we will employ the Taylor expansion theorem to represent properties at z + dz to those at z ... 

The corresponding expression for X. = u is obtained by putting ft = A + e, where e is small, using Taylor s theorem and then letting e tend to zero. We find that... 

The first derivatives in a Taylor expansion, similar to Eqn (28), of the energy E with respect to the occupation numbers rii provide the KS-eigenvalues, as stated by Janak s theorem, and the second derivatives ... 

To prove the Jahn-Teller theorem and to discuss related phenomena we consider the change in the electronic Hamiltonian of a molecule by making small distortions of the nuclei from some chosen origin. For convenience these distortions are represented by vectors in the space of the normal coordinates of the original structure. The change in the Hamiltonian can therefore be written as the Taylor expansion... 

If At 6 E is a function of a real or complex parameter t, we can define differentiation and integration with respect to t, usual rules of operations being applicable to them. Also regularity (analyticity) of At can be defined and Cauchy s, Taylor s and Laurent s theorems are extended to these regular functions. 

On the other hand, the Taylor expansion of Y gives the coefficient ak with de Moivre s theorem in the form48,49 ... 

The proof is given in Appendix A 5.1, where it will become clear that basically the theorem relates the Taylor series expansion of any given function f(x) to the expansion of In /(j ). As such it is a very general statement, of fundamental importance for the thermodynamic limit in any many-body theory. In the present context we note that % definition (Sect. 4.3)... 

Taylor s theorem with remainder, taken to the 1st derivative, is written ... 

Taylor s theorem permits the expansion of certain functions, often in the form of a polynomial. Only the terms which contribute in a significant way to the response are utilized, in this way facilitating the mathematical... 

The topological analysis of p(r, X) then proceeds through the search for and identification of its critical points. In the neighbourhood of a critical point, the field p(r, X) is expanded by Taylor s theorem, the first non-trivial terms being those quadratic in the variables r. The collection of the nine second derivatives of p(r, X) constitute the so-called Hessian matrix A of p(r, X) at the critical point. 

We can apply Taylor s theorem to the left-hand side we also rewrite the right-hand side as... 

With these preliminaries, we can formulate the following Hohenberg-Kohn-type theorem The densities (r, t) and n (r, t) evolving from a common initial state Po = under the influence of two potentials p(r, f) and 0 (both Taylor expandable about the initial time to) are always different provided that the potentials differ by more than a purely time-dependent (r-independent) function ... 

In many practical situations we have to compute a function / (A) of an x TV matrix A. A popular way of computing a matrix function is through the truncated Taylor series approximation. The conditions under which a matrix function / (A) has a Taylor series representation are given by the following theorem (Golub and Van Loan, 1996). 

Consider a bifurcation point Xb for which the charge density p(r X ) exhibits a singularity at of rank co < 3. Then, as a consequence of the above splitting theorem, the charge density may be expanded in a Taylor series in a sufficiently small neighbourhood of r, in which only co components of r appear up to second-order,... 

---