Maclaurin series functions

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 ... 

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

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

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... 

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. 

Sometimes we can generate Maclaurin series for a given function by a simple substitution. For example, the Maclaurin series for the function e can be found as in Problem 1.5 however, an alternative, and much less labour intensive, approach involves writing X= —.x and then using the existing series for e, with X replacing. y. 

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. 

Equation (2.13) is not yet in a form that is fundamentally different from the Cartesian form expressed in equation (2.5). However, we can obtain an alternative, more compact, and far more powerful way of writing the polar form of a complex number by re-visiting the Maclaurin series for the sind, cos6 and exponential functions. The Maclaurin series for cosine and sine are ... 

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. 

Ford, W. B., Studies on Divergent Series and Summability and The Asymptotic Developments of Functions Defined by MacLaurin Series, London Chelsea Publishing, 1960. 

In fact, the proportional error model and exponential error model are equal for small values of a. To see this, the function exp(x) if first written as a MacLaurin series... 

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. 

We also require that all derivatives of the function and of the series be equal at X = 0. This is sufficient for the series to represent the function in some interval around x = 0. Only a function that possesses derivatives of all orders at x = 0 can be represented by a Maclaurin series. Such a function is said to be analytic at X = 0. The nth derivative of the series at x = 0 is... 

This gives us a general formula for the coefficients in a Maclaurin series to represent the function /(x) ... 

A functional series is one way of representing a function. Such a series consists of terms, each one of which is a basis function times a coefficient. A power series uses powers of the independent variable as basis functions and represents a function as a sum of the appropriate linear function, quadratic function, cubic function, etc. We discussed Taylor series, which contain powers of x — h, where h is a constant, and also Maclaurin series, which are Taylor series with h =0. Taylor series can represent a function of x only in a region of convergence centered on h and reaching no further than the closest point at which the function is not analytic. We found the general formula for determining the coefficients of a power series. 

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... 

In the cumulant method (Koppel, 1972 Pusey, J974), the function ln0i(<) is expanded into its Maclaurin series in powers of < at t —> 0 and is written as... 

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. 

For the high temperatures (T Qd), we develop the function to be integrated into a Maclaurin series and integrate, we find ... 

For example, if we wanted to construct a Maclaurin series for the function... 

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