Exponential function Taylor expansion

We will skip the algebraic details. The simple idea is that we can do long division of a function of the form in Eq. (3-30) and match the terms to a Taylor s expansion of the exponential function. If we do, we ll find that the (1/1) Pade approximation is equivalent to a third order Taylor... 

One idea (not that we really do that) is to apply the Taylor series expansion on the exponential function of A, and evaluate the state transition matrix with... 

In a further simplification, namely the expansion of the exponential function in Eq. (2.13) into a Taylor series up to the linear term only (neglecting other terms which may be shown to be negligible) yields... 

Of course, operating on the HF wave function with T is, in essence, full Cl (more accurately, in full Cl one applies 1 + T), so one may legitimately ask what advantage is afforded by the use of the exponential of T in Eq. (7.49). The answer lies in the consequences associated with truncation of T. For instance, let us say that we only want to consider the double excitation operator, i.e., we make the approximation T = T2. In that case, Taylor expansion of the exponential function in Eq. (7.49) gives... 

The model function of a sum of exponentially decaying sinusoids can be approximated by a first-order Taylor expansion. In this way the nonlinear dependence on parameters is approximated by linear dependence on small updates of these parameters.20... 

It is well known that using an exponential or power function can also describe the portion of a polynomial curve. Indeed, these types of functions, which can represent the relationships between the process variables, accept to be developed into a Taylor expansion. This procedure can also be applied to the example of the statistical process modelling given by the general relation (5.3) [5.20]. 

The integral (6.27) can be separated into two parts. In the exponential separation regime, r Taylor expansion. For r > ts the two trajectories separated by a distance comparable to the characteristic lengthscale of the flow are uncorrelated and SS is a stationary chaotic function, thus... 

Analogous to the Taylor series expansion of an exponential function, if a matrix appears as the argument of an exponential operator, then the function is expanded as follows ... 

In the case of a thin absorber (ta —> 0), the Taylor expansion of the exponential and Bessel functions yields a proportional relationship between the peak area and the number of resonant... 

The Taylor expansion (Mathematical toolkit 3.2) of an exponential function used in Justification 13.1 is... 

Several major generalizations are needed to discuss vibrational and NLO spectra. The CH units represented by M in Fig. 6.1 have vibrational degrees of freedom, and the PA backbone is planar rather than one-dimensional. The amplitude mode (AM) formalism developed by Horovitz and coworkers [18,19] extends the SSH model to several coupled q = 0 modes, as discussed in Section II.C, with the partitioning taken from experiment. The second term in the Taylor expansion of t(R) appears in force fields, as already recognized by Coulson and Longuet-Higgins [20]. While the form of t (R) need not be specified in advance, wave function overlap is usually taken to be exponential, and this fixes the curvature without additional parameters as... 

The value of y limits to unity for rather high surface potentials. The Hamaker constant, A, must be inserted in J, CCC in mol and z is the counter-ion valency (including sign). Thus, in this case, CCC is proportional to 1/z . At low potentials, y limits to zey/o/ AksT (use of Taylor expansion of exponential function followed by This leads to CCC being... 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

The Taylor series expansion in Chapter 2 makes it possible to derive a remarkable relationship between exponentials and trigonometric functions, first found by Euler ... 

The current density presented in Figure 5.4(b) cem be described as being a linear function of potential over a narrow range of potential near the zero-current potential. Taylor series expansions of the exponential terms in equation (5.14) yield... 

Solving the above system of equations the coefficients of the methods are obtained. For small values of w the formulae obtained by the solution of the system of equations are subject to heavy cancellations. In this case Taylor series expansions must be used. The corresponding exponentially fitted method (i.e. a method which integrates exactly any linear combination of the functions 1, X, x, x , x, exp(ihwx) ) is produced via the substitution w = —iq>. 

The exponential form of Eq. 6-28 is a nonrational transfer function that cannot be expressed as a rational function, a ratio of two polynominals in s. Consequently, (6-28) cannot be factored into poles and zeros, a convenient form for analysis, as discussed in Section 6.1. However, it is possible to approximate by polynomials using either a Taylor series expansion or a Fade approximation. 

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