Maclaurin series expansion

For film diffusion, but not particle diffusion, the rate of exchange is directly dependent upon the total external concentration. Consider a very small degree of exchange such that F( ) 1 in which case the Maclaurin series expansion gives log (1 — F) = — F, and the boundary conditions more nearly approach those for trace exchange (infinite volume). In such a case, equation 6.19 may be written ... 

I) = 5, it is excluded from the domain of the series. We can further illustrate this by comparing a plot of the function y = f x) = (1 - x)" with the MacLaurin series expansion of this function up to the third, fourth, fifth and sixth terms (see Figure 1.5). Clearly the three plots match quite well for — 1 <. < 1 but differ dramatically for all other values of -V. We also see at x = -1 that the series representation oscillates between zero and +1 as each new term is added to the series, thus indicating divergence at this point. 

Here we have solved successively, the first, second and third equations, sequentially, and then in reverse order to make a symmetric (and second order) integrator. The auxiliary function involved c t, = Lz l is easily evaluated from a few terms of its Maclaurin series expansion... 

This form has the advantage of not containing concentration value m and thus permits the explicit expression of x. from the basic equation. The expansion of the general function given by "Equation 17 into the MacLaurin series with respects to molalities m yields the equation ... 

In the following, we show that the coefficients a , in Eq. (3.31) are related to the derivatives of the sample wavefunction i ) with respect to X, y, and z at the nucleus of the apex atom in an extremely simple way. (To simplify the notation, we take the nucleus of the apex atom as the origin of the coordinate system, i.e., xo = 0, yo = 0, and zo - 0.) This is similar to the well-known case that the expansion coefficients for a power series are simply related to the derivatives of the function at the point of expansion, the so-called Taylor series or MacLaurin series. We will then obtain the derivative rule again, from a completely different point of view. 

Assuming that the perturbation is not too large we may expand fl (T, /x A) in a MacLaurin series in terms of the coupling parameter A, that is, as a Taylor expansion in A around A = 0 (i.e., the reference system). Retaining in this expansion terms up to first order, we obtain... 

Understand the distinction between Maclaurin and Taylor series expansions and appreciate when one or the other is the more appropriate... 

For any power series expansion, the accuracy of a polynomial truncation depends upon the number of terms included in the expansion. Since it is impractical to include an infinite number of terms (at which point the precision is perfect), a compromise has to be made in choosing a sufficient number of terms to achieve the desired accuracy. However, in truncating a Maclaurin series, the chosen degree of polynomial is always going to best represent the function close to x = 0. The further away from x = 0, the worse the approximation becomes, and more terms are needed to compensate, a feature which is demonstrated nicely in Figure 1.2 and Table 1.2. 

Figure 1.3 An illustration of the improved accuracy achieved with the Taylor series expansion of f x)=e about x=1, compared with the Maclaurin series (expanded about x=0)...

Regardless of the signs of x or a, e" approaches 1, for increasingly small values of x, according to the MacLaurin power series expansion (as seen in Problem 1.9a) ... 

It is readily shown that a defines a Temperature scale.) The theory has thus derived these "laws" as truncated expansions of a Maclaurin series, and shows that transport properties are dependent upon (a,p). The theory also predicts the existence of (a,p) dependence for u,b, and v. However, the functions themselves must be determined via experiment. The methods for establishing these functions from experimentally measurable functions are not a part of this paper, inasmuch as they follow the lines presented in the typical presentations of thermodynamics. 

If the series is expanded around zero, the expansion is known as a MacLaurin series. The approximation consists of a linear combination of the function evaluated at x0 and derivatives of increasing degree. 

When a Taylor expansion of/(jc) is carried out about jc = 0, it is called a Maclaurin series. The general formula is... 

A.7 Obtain a Taylor expansion of exp(ajc) about ax =. Obtain the Maclaurin series. What approximations could be made for exp(ax) near ax = I and ax = 01... 

Otherwise we may fall back upon Maclaurin s expansion in ascending powers of a , the constants being positive, negative or zero. This series is particularly useful when the terms converge rapidly. When the results exhibit a periodicity, as in the ebb and flow of tides annual variations of temperature and pressure of the atmosphere cyclic variations in magnetic declination, etc., we refer the results to a trigonometrical series as indicated in the chapter on Fourier s SBries. - ... 

Concurrently with the above mathematical treatment there exists cinother approach to the problem of the determination of the expUcit form of the function t(AE) starting from equation (11), which is based on its MacLaurin s series expansion [57]... 

A Maclaurin series is a specific form of the Taylor series for which Xq = 0. Some standard expansions in Taylor series form are ... 

This result is known as Taylor s theorem, and the expansion is a Taylor series. The case xq = 0, given by Eq. (7.40), is sometimes called a Maclaurin series. 

Since the Thomas-Fermi functional is exact for the uniform electron gas, its failings must arise because the electron densities of chemical substances are far from uniform. This suggests that we construct the gradient expansion about the uniform electron gas limit such functionals will be exact for nearly uniform electron gases. An alternative perspective is to recall that the Thomas-Fermi theory is exact in the classical high-quantum number limit. The gradient expansion can be derived as a Maclaurin series in powers of ti it adds additional quantum effects to the Thomas-Fermi model. 

To find another useful relationship between sin 6 and cos 0 in the complex plane, let us expand each function in terms of a Maclaurin series. (Series expansions are covered in detail in Chapter 7.)... 

A Maclaurin s series expansion in density around p = 0 yields ... 

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