Series Maclaurin

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. 

A common situation in thermodynamics is that some property z(x) and its lower derivatives (zf, z", z ",. ..) have been measured at a certain point x0, and one wishes to use this information to approximate the behavior of the function z(x0 + Ax) in the Ax-neighborhood of x0. For this purpose, the fundamental Taylor series (or MacLaurin series, the special case for xo = 0) yields approximations that are useful for sufficiently small Ax ... 

We will denote the solvent by the subscript 0. Usually In t/ is expressed as a Maclaurin series in which only the first term is retained—i.e.,... 

This is an expression of the fact that at constant heat flux the melt thickness initially grows linearly with time. For small times an exact solution of this problem has been given by Evans et al. (E3), who expanded Ai (tji) in a Maclaurin series and found the first five coefficients by direct substitution ... 

Table A1.3. Maclaurin series for some simple functions...

Here, as usual, the remainder symbol 0(u2K) represents a Maclaurin series in powers of u such that the starting term is u1K. 

The Thomas precession frequency coj is defined as the difference between 2%/T and 2%/T, where we keep only the first two terms of a Maclaurin series ... 

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 ... 

An analogical - but empirical - equation was given earlier by Szyszkowski (1908). When the surface tension of both the components does not differ much, Eq. (6.34) leads to a simple additivity law for surface tensions. If ai and a2 are sufficiently close, then the exponentials can be expanded into a MacLaurin series giving... 

If we recall from calculus that many functions can be written as a Maclaurin series, then... 

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... 

This latter expression can be derived independently by expanding in the mean-field equation of state (see Eq. (4.28)] the term 1/(1 — bp) bp 1) in a MacLaurin series according to... 

The mentioned uniqueness of the Fade approximant for the given input Maclaurin series (4) represents a critical feature of this method. In other words, the ambiguities encountered in other mathematical modelings are eliminated from the outset already at the level of the definition of the PA. Moreover, this definition contains its "figure of merit" by revealing how well the PA can really describe the function G(z ) to be approximated. More precisely, given the infinite sum G(2 ) via Eq. (4), the key question to raise is about the best agreement between from Eq. (2) and G(z ) from... 

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