Series Cosine

For the Fourier coefficients of the modulus and the phase we note that, because of the time-inversion invariance of the amplitude, the former is even in f and the latter is odd. Therefore the former is representable as a cosine series and the latter as a sine series. Formally,... 

Many of the torsional terms in the AMBER force field contain just one term from the cosine series expansion, but for some bonds it was found necessary to include more than one term. For example, to correctly model the tendency of O-C—C-O bonds to adopt a gauche conformation, a torsional potential with two terms was used for the O—C—C—O contribution ... 

In case the curve y = fix) is symmetrical with respect to the origin, the a s are all zero, and the series is a sine series. In case the curve is symmetrical with respect to the y axis, the fc s are all zero, and a cosine series results. (In this case, the series will be valid not only for values of x between — c and c, but also for x = — c and x = c.) A Fourier series can always be integrated term by term but the result of differentiating term by term may not be a convergent series. 

The Fourier expansion (B.l) may also be expressed as a cosine series or as a sine series by the introduction of phase angles a ... 

Without repeating the work, a similar result is obtained for S(x) = cos m. Combination of sine and cosine series leads to the even more general exponential form, such that... 

The price for regularization is one additional differential equation. An approximate way of regularization which results in only two equations of motion is obtained in the following way. Since according to (6.1.18) x is a periodic function of tj, which itself is a periodic function of 6, the variable x can be expanded in a Fomrier series. Following Landau and Lifschitz (1977), we expand the position variable x I,0) in (6.1.18) into a Fomrier-cosine series ... 

V and are periodic functions of t and can be represented in terms of a Fourier series. Because of the intrinsic symmetry of the dihedral potential, as well as the symmetry of ethane, only the cosine series exists in this case (see the discussion for propane below ), i.e.,... 

Finally, a note on disorder of the membrane stacks and on attempts to correct for it in the analysis of diffraction data. Generally, two kinds of disorder are being discussed in crystal structure Disorder of the first kind refers to displacements of the structural elements (for example the one-dimensional unit cell of a membrane stack) from the ideal positions prescribed by the periodic lattice. The effect on the diffraction pattern is indistinguishable from that of thermal vibrations and may, therefore, be expressed as a Debye-Waller temperature factor so that the structure factor, expressed as a cosine series, includes a Gaussian terra, according to... 

If the function f x) is an even function, all of the bn coefficients will vanish, and only the cosine terms will appear in the series. Such a series is called a Fourier cosine series. If f x) is an odd function, only the sine terms will appear, and the series is called a Fourier sine series. If we want to represent a function only in the interval 0 < a < L we can regard it as the right half of an odd function or the right half of an even function, and can therefore represent it either with a sine series or a cosine series. These two series would have the same value in the interval 0 < X < L but would be the negatives of each other in the interval — L < jc < 0. 

By reference to page 283, we shall find that the first expression in brackets, is the cosine series, the second the sine series. Hence,... 

By plotting the first three terms enclosed in brackets on the right side of (9), we obtain the series of curves shown in Fig. 159. The general development of a cosine series is... 

IL Comparison of the sine and the cosine series. The sine and cosine series are both periodic functions of x, with a period of %r. The above expansions hold good only between the limits x = + 7r, that is to say, when x is greater than - 7r, and less than + ir. When x = 0, the series is necessarily zero, whatever be the value of the function. As a matter of fact any function can be represented both as a sine and as a cosine series. Although the functions and the two series will be equal for all values of x between x — tt and x = 0, there is a difference between... 

Not all molecular mechanics force fields use torsional potentials it may be possible to rely upon non-bonded interactions between the atoms at the end of each torsion angle (the 1,4 atoms) to achieve the desired energy profile. However, most force fields for orgamc molecules do use explicit torsional potentials with a contribution from each bonded quartet of atoms A-B—C-D in the system. Thus there would be nine individual torsional terms for ethane and 24 for benzene (6 x C—C—C—C, 12 x C—C—C—H and 6 x H—C—C—H). Torsional potentials are almost always expressed as a cosine series expansion. One functional form is ... 

The series representing fix) is called a Fourier-Sine series. Similarly, we can also express functions in terms of the Fourier-Cosine series... 

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