The use of Fourier series

G.7 R. W. James. The Crystalline State. Vol. 11 The Optical Principles of the Diffraction of X-Rays (London George Bell, 1948). Excellent book on advanced theory of x-ray diffraction. Includes thorough treatments of diffuse scattering (due to thermal agitation, small particle size, crystal imperfections, etc.), the use of Fourier series in structure analysis, and scattering by gases, liquids, and amorphous solids. 

This relation and equation [5.107] serve as the basis for the use of Fourier series analysis of diffraction peak profiles to quantitatively characterize stmctural defects found in crystals. We will see in the following chapter how this result can help us determine, in particular, the size and microstrain distribution functions. 

One of the most common solution techniques applicable to linear homogeneous partial differential equation problems involves the use of Fourier series. A discussion of the methods of solution of linear partial differential equations will be the topic of the next chapter. In this chapter, a brief outline of Fourier series is given. The primary concerns in this chapter are to determine when a function has a Fourier series expansion and then, does the series converge to the function for which the expansion was assumed Also, the topic of Fourier transforms will be briefly introduced, as it can also provide an alternative approach to solve certain types of linear partial differential equations. 

In developing the differential equation for circular plates, the shearing stress was ignored because the load was symmetric with respect to 6. In rectangular plates under uniform loads, the shearing stress interacts with the normal stresses in the X- and y-directions and thus cannot be ignored. This results in a more complicated differential equation than that for circular plates. In addition, the solution of the differential equation of rectangular plates is more elaborate and involves the use of Fourier series. Because of this, only the case of a simply supported... 

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. 

P 7] The topic has only been treated theoretically so far [28], A mathematical model was set up slip boundary conditions were used and the Navier-Stokes equation was solved to obtain two-dimensional electroosmotic flows for various distributions of the C, potential. The flow field was determined analytically using a Fourier series to allow one tracking of passive tracer particles for flow visualization. It was chosen to study the asymptotic behavior of the series components to overcome the limits of Fourier series with regard to slow convergence. In this way, with only a few terms highly accurate solutions are yielded. Then, alternation between two flow fields is used to induce chaotic advection. This is achieved by periodic alteration of the electrodes potentials. 

There are problems in determining crystallite size from line broadening alone, since factors other than crystallite size contribute to the broadening, including local strain in the crystallites and shape anisotropy. Some of these problems can be overcome by the use of Fourier analysis of the peak shape. The cosine coefficients of the Fourier series can be used to determine a surface weighted average size for the crystallites. 

A more mathematical approach would invoke a theorem of differential equations, which says that a second order partial differential equation that is as nice as the one we have here, with two initial conditions of the form we just used, must have a unique solution. The branch of mathematics you would have to study to learn this theorem is called partial differential equations sometimes, or if the professor plans to give you the most general version, the area of study might be called differential operators on manifolds. Mathematically, this type of theorem makes the claim for our model of nature that the physical explanation is attempting to make for nature herself. Then we would again invoke the theory of Fourier series to tell us that the sines and cosines are good enough to do the job. 

A whole series of orthonormal functions can be used to interpret the information. The most familiar and applicable are the Fourier functions. Before being able to compose a particle shape descriptor in the polar system by the use of Fourier functions, one must realize that all that is normally known of a particle is its silhouette or profile. Therefore, methods must be found to interpret information from cuts through the particle or scans of portions of the surface area and connect it with overall shape. It is assumed that the silhouette of any cut or sample of the surface will give all information, such as roughness and other physical parameters, needed to describe the entire particle surface. Thus, unless the silhouette of a particle misses a unique, dominant feature of the particle shape, it will be representative of the particle. By sampling... 

An adequate discussion of the conditions of convergency of Fourier s series must be omitted. W. E. Byerly s An Elementary Treatise on Fourier s Series, etc., is one of the best practical works on the use of Fourier s integrals in mathematical physics. J. Fourier s pioneer work Thiorie analytiquc de la Chaleur, Paris, 1822, is perhaps as modern as any other work on this subject see also W. Williams, Phil. Mag. [5], 42, i25,1896 Lord Kelvin s Collected Papers and Riemann-Weber s work (Z.c.), etc. 

Following are the four examples of the expediency that is achievable when the concept of odd or even function is used in the context of Fourier series. 

These symmetry relations will be used to advantage in the application of Fourier-series techniques to the description of periodic functions. 

Solutions to Fourier s equation are in the form of infinite series but are often more conveniently expressed in graphical form. In the solution the following dimensionless groups are used. 

Though he won the prize he did not win the outright acclaim of his referees. They accepted that Fourier had formulated heat flow correctly but felt that his methods were not without their difficulties. The use of the Fourier Series was still controversial. It was only when he had returned to Paris for good (around 1818) that he could get his work published in his seminal book, The Analycical Theoiy of Heat. 

Since the article by Spedding1 on infrared spectroscopy and carbohydrate chemistry was published in this Series in 1964, important advances in both infrared and Raman spectroscopy have been achieved. The discovery2 of the fast Fourier transform (f.F.t.) algorithm in 1965 revitalized the field of infrared spectroscopy. The use of the f.F.t., and the introduction of efficient minicomputers, permitted the development of a new generation of infrared instruments called Fourier-transform infrared (F.t.-i.r.) spectrophotometers. The development of F.t.-i.r. spectroscopy resulted in the setting up of the software necessary to undertake signal averaging, and perform the mathematical manipulation of the spectral data in order to extract the maximum of information from the spectra.3... 

The classical approach to the analysis of mixtures by use of infrared spectroscopy consists in identifying specific, strong bands that belong to a suspected component, obtain a pure spectrum of the suspected component, and then remove those in the spectrum of the mixture that are due to the identified compound. The process is repeated for the remaining bands in the mixture spectra. Once the component spectra are known for a mixture, a series of calibration curves is produced. These curves relate concentration to absorbance, using Beer s law. The concentration of the components of the mixture are then obtained by interpolation. The advantage of Fourier-transform, infrared spectroscopy is that components of a mixture may be... 

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