Power series Maclaurin

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. 

An alternative way of computing the coefficients [p , -r and q , -r comes from observing that the two power series representations in Eq. (114) are the truncated Maclaurin expansions. Thus,... 

The coefficients, a, are often related to one another in a simple way which is determined by the nature of the function. An important method of expressing functions in a power series is the Taylor and Maclaurin expansions. In a Taylor expansion the function f(x) is expanded about a given point xq and the coefficients are related to the values of the derivatives of the function at x = xq. Thus, the Taylor expansion of f(x) is... 

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. 

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) ... 

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. 

A different approach consists of representing the real equation of state by MacLaurin s power series. In practice, there are two virial equations, one describing the product PV as a function of the reciprocal of the molar volume and the second using the variable P. 

Under isothermal conditions, the absolute gas pressure can be expressed as a function of the reciprocal of the molar volume P = fil/VJ =RT/V, and the developed MacLaurin s power series of this equation is as follows ... 

The most common type of functional series is the power series, which uses powers of the independent variable as basis functions. The first type of power series is the Maclaurin series ... 

Infinite series, 94-109 comparison, 95 convergence of, 94-99 tests for, 94-99 definition of, 94 divergence of, 94-99 tests for, 94-99 Fourier, 101-106 Maclaurin, 99 power. See Power series Taylor, 99... 

In this chapter we consider several methods of expanding functions in infinite series. Two, which are particularly useful in physical chemistry, are the power series known as the Maclaurin series and the Taylor series. Let us consider the Maclaurin series first. Suppose that a function y(x) can be expanded in a power series... 

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

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 ... 

Maclaurin s theorem determines the law for the expansion of a function of a single variable in a series of ascending powers of that variable. Let the variable be denoted by x, then,... 

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. - ... 

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. 

Next, we expand the LHS and the exponential of the integrand in the RHS of Eq. 8.81 in Maclaurin series (about t = 0) and equate terms with the same power, thus getting... 

A Maclaurin series is a functional series with basis functions that are power ofx, where jc is the independent variable. 

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