Fourier series functions

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Consider a periodic function x(t) that repeats between t = —r/2 and f = +r/2 (i.e. has period t). Even though x t) may not correspond to an analytical expression it can be written as the superposition of simple sine and cosine fimctions or Fourier series, Figure 1.13. 

Hg. 1.14 The connection between the Fourier transform and the Fourier series can be established by gradually increasing the period of the function. When the period is infinite a continuous spectrum is obtained. (Figure adapted from Ramirez R W, 1985, The FFT Fundamentals and Concepts. Englewood Cliffs, NJ, Prenhce Hall.)... 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

The coefficients C are chosen to ensure that the function has a minimum at the appropriate reference bond angle. For linear, trigonal, square planar and octahedral coordination, Fourier series with just two terms are used with a Cq term and a term for n = 1, 2, 3 or 4, respectively ... 

Hence we see that this simple periodic function has just two terms in its Fourier series. In terms of the Sine and Cosine expansion, one finds for this same f(t)=Sin3t that an = 0, bn =... 

Earlier, for Fourier series, we had the orthogonality relation among the Fourier functions ... 

If f is a function of several spatial coordinates and/or time, one can Fourier transform (or express as Fourier series) simultaneously in as many variables as one wishes. You can even Fourier transform in some variables, expand in Fourier series in others, and not transform in another set of variables. It all depends on whether the functions are periodic or not, and whether you can solve the problem more easily after you have transformed it. 

The only limitation on the function expressed is that it has to be a function that has the same boundary properties and depends on the same variables as the basis. You would not want to use Fourier series to express a function that is not periodic, nor would you want to express a three-dimensional vector using a two-dimensional or four-dimensional basis. 

In molecular mechanics, the dihedral potential function is often implemented as a truncated Fourier series. This periodic function (equation 10) is appropriate for the torsional potential. 

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

The original idea of approximating the quantum mechanical partition function by a classical one belongs to Feynman [Feynman and Vernon 1963 Feynman and Kleinert 1986]. Expanding an arbitrary /S-periodic orbit, entering into the partition-function path integral, in a Fourier series in Matsubara frequencies v . 

One other approach is to measure the rate as a function of a specified argument alone, and then differentiate the function with respect to the argument used. The differentiation can be done graphically, or by fitting an empirical function to the data (like a Fourier series) and differentiating this analytically. 

The breakdown of a given signal into a sum of oscillatory functions is accomplished by application of Fourier series techniques or by Fourier transforms. For a periodic function F t) with a period t, a Fourier series may be expressed as... 

Table 2.3 Comparison of functional forms used in common force fields. The torsional energy, [ors is in all cases given as a Fourier series in the torsional angle...

Table 2.3 gives a description of the functional form used in some of the common force fields. The torsional energy is written as a Fourier series, typically of order three, in all cases. Many of the force fields undergo developments, and the number of atom types increases as more and more systems become parameterized thus Table 2.3 may be considered as a snapshot of the situation when the data were collected. The universal type force fields, described in Section 2.3.3, are in principle capable of covering molecules composed of elements from the whole periodic table, these have been labelled as all elements . 

The above equation is known as a Fourier series, which is a function of time or /(/). The amplitudes (A , A2, etc.) of the various discrete vibrations and their phase angles (cpi, cj>2, 3 ) can be determined mathematically when... 

Mathematical theory shows that any periodic function of time, /(f), can be represented as a series of sine functions having frequencies a>, 2a>, 3ft), 4ft), etc. Function /(f) is represented by the following equation, which is referred to as a Fourier series ... 

We may solve for the electron distribution function by expanding it in Legendre polynomials in cos 6 (where v = (v,6,Fourier series in cot we shall use here only the first-order terms ... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

This analysis is based on the fact that any periodic function can be approximated, at any order, by a Fourier series ... 

Obviously, the state function (jc) is not an eigenfunction of H. Following the general procedure described above, we expand in terms of the eigenfunctions n). This expansion is the same as an expansion in a Fourier series, as described in Appendix B. As a shortcut we may use equations (A.39) and (A. 40) to obtain the identity... 

A similar expression applies for b . The generalization for a function /(9) with a finite number of finite discontinuities is straightforward. At an angle 9q of discontinuity, the Fourier series converges to a value of /(9) mid-way between the left and right values... 

Section 3.4. If some of the terms in the Fourier series are missing, so that the set of basis functions in the expansion is incomplete, then the corresponding coefficients on the right-hand side of equation (B.16) will also be missing and the equality will not hold. 

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