Fourier expansions

In the case of ethylene, because of 2-fold symmetry, odd terms drop out of the series, V3, V5,... = 0. In the case of ethane, because of 3-fold symmeti-y, even temis drop out, V2, V4,... = 0. Terms higher than three, even though permitted by symmetry, are usually quite small and force fields can often be limited to three torsional terms. Like cubic and quaitic terms modifying the basic quadratic approximation for stretching and bending, terms in the Fourier expansion of Ftors (to) beyond n = 3 have limited use in special cases, for example, in problems involving octahedrally bound complexes. In most cases we are left with the simple expression... 

Oberhettinger, F. Fourier Expansions A Collection of Formulas, Academic, New York (1973). 

The solution to the governing differential equation, Equation (5.32), is not as simple as for specially orthotropic laminated plates because of the presence of D. g and D2g. The Fourier expansion of the deflection w. Equation (5.29), is an example of separation of variables. However, because of the terms involving D.,g and D2g, the expansion does not satisfy the governing differential equation because the variables are not separable. Moreover, the deflection expansion also does not satisfy the boundary conditions. Equation (5.33), again because of the terms involving D. g and D2g. 

Ashton solved this problem approximately by recognizing that the differential equation, Equation (5.32), is but one result of the equilibrium requirement of making the total potential energy of the mechanical system stationary relative to the independent variable w [5-9]. An alternative method is to express the total potential energy in terms of the deflections and their derivatives. Specifically, Ashton approximated the deflection by the Fourier expansion in Equation (5.29) and substituted it in the expression for the total potential energy, V ... 

In both potentials in eqs. (2.27) and (2.28) the bairier towards linearity is given implicitly by the force constant. A more general expression which allows even quite complicated energy functionals to be fitted is a general Fourier expansion. 

In Taylor expansions, Fourier expansions, lagrangean or newtonian... 

Inserting the perturbation and Fourier expansion of the cluster amplitudes and the Lagrangian multipliers,... 

There is an alternative - and for our purposes more powerful - way to estimate the discretization error, namely in terms of the Fourier expansion of a periodic -function. We write hf xk), see (A.l), as [24]... 

Fortunately the Fourier expansion method helps us for small and intermediate h but large n. We get in the limit n - oo... 

The Fourier expansion of the density in an extended system which does not have any particular symmetry is... 

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 ... 

Fourier expansion of p10(a) allows us to calculate relative intensities of the three forbidden bands m2, m, and m,. These are in quantitative agreement with experiment. The agreement is excellent over a wide range of ratios of the key model parameters V6 and F (effective rotational constant), which are taken from experiment. Previous conformational inferences in S0, S, D0 (ground state cation), including our own, were in fact correct. They now rest on solid ground. 

By substituting the Fourier expansions of the field into this expression... 

Until now, our analysis has been purely formal and it is just for mathematical convenience that we have performed the Fourier expansion (39) and (40) for pN. However, we shall see here that this procedure has a very intuitive meaning, which has far-reaching consequences. 

When combined with the Fourier expansion of functions, separation of variables is another powerful method of solutions which is particularly useful for systems of finite dimensions. Regardless of boundary conditions, we decompose the solution C(x, t), where the dependence of C on x and t is temporarily emphasized, to the general one-dimensional diffusion equation with constant diffusion coefficient... 

Such a sine expansion is generally made possible with the condition of zero concentration at x=0 by using an odd function for the initial concentration distribution. A simple example is for initial uniform concentration C0 between 0 and X for which we can assume a fictitious concentration — C0 between — X and 0. Using the results of Chapter 2, the Fourier expansion of the boxcar function which is C0 between 0 and X, and 0 at x=0 and x = X is... 

The function that has such a Fourier expansion while satisfying C(0,f)=C(0,. Y)=0 is the boxcar function. We found in Section 2.6.2 that the a coefficients of this function are 0 for even values and 4/nn for odd values of n. The general solution is therefore... 

In order to make the solution consistent with initial and boundary conditions, we will use for u(r,0) the ramp function defined in Chapter 2. For 0[Pg.447]

Stacking fault contrast in section topographs can be most easily understood by expanding the incident spherical wave as a Fourier expansion of plane waves... 

Compositional fluctuations may be represented through a Fourier expansion, and are sufficiently approximated (for our interests) by the first sinusoidal term... 

As we have discussed previously, any function with two-dimensional periodicity can be expanded into two-dimensional Fourier series. If a function has additional symmetry other than translational, then some of the terms in the Fourier expansion vanish, and some nonvanishing Fourier coefficients equal each other. The number of independent parameters is then reduced. In general, the form of a quantity periodic in x and y would be... 

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