Maclaurin series, Taylor expansions

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. 

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... 

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... 

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)...

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... 

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. 

---