Fourier series expansion

Besides the intrinsic usefulness of Fourier series and Fourier transforms for chemists (e.g., in FTIR spectroscopy), we have developed these ideas to illustrate a point that is important in quantum chemistry. Much of quantum chemistry is involved with basis sets and expansions. This has nothing in particular to do with quantum mechanics. Any time one is dealing with linear differential equations like those that govern light (e.g. spectroscopy) or matter (e.g. molecules), the solution can be written as linear combinations of complete sets of solutions. 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

Equations. For a ID two-phase structure Porod s law is easily deduced. Then the corresponding relations for 2D- and 3D-structures follow from the result. The ID structure is of practical relevance in the study of fibers [16,139], because it reflects size and correlation of domains in fiber direction . Therefore this basic relation is presented here. Let er be50 the direction of interest (e.g., the fiber direction), then the linear series expansion of the slice r7(r)]er of the corresponding correlation function is considered. After double derivation the ID Fourier transform converts the slice into a projection / Cr of the scattering intensity and Porod s law... 

Changing x to — x in the first integral and recognizing that cos is an even function shows that b is zero for all n. The Fourier series expansion of the boxcar function is therefore... 

Find the Fourier series expansion of the ramp function (Figure 2.11) defined over the interval [—X,X] by... 

It will be left as an exercise for the reader to show that the cosine terms bn, including the mean value b0, are zero. The Fourier series expansion of the ramp function is therefore... 

Here cp represents the torsional coordinate, I denotes the moment of inertia, and Vf is the first coefficient of the Fourier series expansion of the isomerization potential of periodicity nn. 

Hence, h s are the coefficients of the Fourier sine series expansion of Cq —Ci. The values of h s can be obtained by multiplying both sides by sm. mnx/L) and integrating from 0 to L using the following relationship ... 

Because of the observed asymmetry in the anti-peak, Eqs. (13) and (14) were applied over a region corresponding to 2o

determined values of AH, torsional potential was estimated including V[-, V2-, V3-terms in the Fourier series expansion. The resulting... 

The envelope e(n) of the resulting signal in general is not equal to the original envelope a(n) but will follow a(n) due to the periodicity of the second term of e(n) in Equation (9.95). Now at this point we could assume a model for the phase (])( ) in the form of a nested modulator with a resulting phase residual. An alternative, as argued by Justice [Justice, 1979], is to note that ( )(n) is a periodic function and express it as a Fourier series expansion i.e.,... 

For large amplitude co-ordinates involving inversion or internal rotation a power series expansion in the displacements may not be appropriate (for example, a Fourier expansion is clearly preferable for internal rotation), and the force constants will then be defined in a different way. However, such examples are not considered here. 

They have been designed by the method of simulated annealing and further refinement procedures [Geel]. The pulse shape is represented by a Fourier series expansion. 

The Fourier series expansions of a function /(x) of the variable x over the range —I X I may be generalized to the case where the range is infinite, i.e., where... 

If h y(r) is analytic, and decays fast enough so that the integral exists, the Fourier transform is an even function of k. Further, sin(fcr) has an expansion in odd powers of k and thus h y(k) will have a power series expansion in even powers of k ... 

A number of methods have been proposed for particle shape analysis these include verbal description, various shape coefficients and shape factors, curvature signatures, moment invariants, solid shape descriptors, the octal chain code and mathematical functions like Fourier series expansion or fractal dimensions. As in particle size analysis, here one can also detect intense preoccupation with very detailed and accurate description of particle shape, and yet efforts to relate the shape-describing parameters to powder bulk behaviour are relatively scarce.10... 

The stability attributes of the aforementioned method are explored via the combined von Neumann and Routh-Hurwitz technique [2]. According to this approach, the error that appears during the computation of any field component is described by a single term of a Fourier series expansion ... 

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