Taylor series exponential

This is simply a special case of Taylors series when h is set to zero. Exponential Series... 

To evaluate L ip) from equation (F.2), we first factor out in the generating function and expand the remaining exponential function in a Taylor series... 

To handle the time delay, we do not simply expand the exponential function as a Taylor series. We use the so-called Pade approximation, which puts the function as a ratio of two polynomials. The simplest is the first order (1/1) Pade approximation ... 

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... 

With little error the exponential in the numerator of the rightmost fraction of Eq. (7.2) is expanded in a Taylor series resulting in... 

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... 

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... 

Selected entries from Methods in Enzymology [vol, page(s)] Claverie approach to Pade-Laplace algorithm for sums of exponentials, 210, 59 Taylor series expansion in analysis of sums of exponentials, 210, 56. 

Since the susceptibility cannot be evaluated in a deterministic way, the exponential function is developed at p = 0 (which implies T —> oo) as a Taylor series ... 

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 nonlinearity in Eqs. (3.9)—(3.11) occurs in the product of variables and in the exponential temperature term. Expanding these nonlinear terms in a Taylor series and truncating after the first term give three linear ordinary differential equations ... 

These results reveal that the same law describes release from both fractal and Euclidean matrices. The release rate is given by the time derivative of (4.14). For early stages of the release, calculating the derivative of (4.14) and performing a Taylor series expansion of the exponential will result in a power law for the... 

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... 

The concentration profile for vanishing convection h —> 0 as given in (1.187) is obtained from (1.190). A Taylor series of the exponential function at h = 0 can be developed to indicate this. The material flux transferred to the surface y = 0 (index 0 = wall) in the y-direction is... 

If we assume that the ions are stationary (a good approximation for say a radio-frequency discharge where because of their large mass the ions will not respond to the rapidly changing field), then n+ = n0, and deep in the plasma where e /kTe is small the exponential can be expanded in a Taylor series. Equation 49 then becomes... 

Using the definition above, why is a system with dead time a higher-order system (Hint Consider the Taylor series expansion of an exponential term. See also item 4.)... 

The exponential term can then be expanded in a Taylor series. If we collect the lowest-order terms of Dp, we then have... 

In the limit of low vibration frequencies, when hv < kT, the exponentials can be expanded into a Taylor series and the vibration partition function is simplified to... 

If the resist is only slightly absorbing, such that a-yD Taylor series expansion as... 

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 ... 

The function B z Zc), which can be expressed in terms of an exponential integral, has the behavior we expect for autoionization in that it has a branch point at Zc at which the function becomes complex but with the derivative continuous along the real axis. The coefficients of the polynomials can be chosen so that the Taylor series agrees with the perturbation series. 

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