Trigonometric Representation

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

So far we have seen that if we begin with the Boltzmann superposition integral and include in that expression a mathematical representation for the stress or strain we apply, it is possible to derive a relationship between the instrumental response and the properties of the material. For an oscillating strain the problem can be solved either using complex number theory or simple trigonometric functions for the deformation applied. Suppose we apply a strain described by a sine wave ... 

Like PCA compression, Fourier compression involves the reduction of an analyzer response profile into a simpler representation that uses basis functions that are linear combinations of the original variables. However, in the case of Fourier compression, these basis functions are pre-defined trigonometric functions of the original variables, whereas in PCA they (the PC loadings) are application-specific and must be determined through an analysis of the entire data set. 

Table 14 Superimposition of the pyrocatechin G NRG (capital letters) and the Civ symmetry point (small letters) character tables, with their representations in trigonometric basis sets, as well as the electric dipole moment components...

In the same table the representations of the G4 and Cqu groups, in trigonometric basis sets are given, as well as the electric dipole moment components. So, it is seen that the z, x and y electric dipole components transform according the /li, Ai and representations of the G4 group, respectively. 

Any periodic function (such as the electron density in a crystal which repeats from unit cell to unit cell) can be represented as the sum of cosine (and sine) functions of appropriate amplitudes, phases, and periodicities (frequencies). This theorem was introduced in 1807 by Baron Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist who pioneered, as a result of his interest in a mathematical theory of heat conduction, the representation of periodic functions by trigonometric series. Fourier showed that a continuous periodic function can be described in terms of the simpler component cosine (or sine) functions (a Fourier series). A Fourier analysis is the mathematical process of dissecting a periodic function into its simpler component cosine waves, thus showing how the periodic function might have been been put together. A simple... 

Firstly, they provide a means to formulate alternative representations of transcendental functions such as the exponential, logarithm and trigonometric functions introduced in Chapter 2 of Volume 1. Secondly, as a direct result of the above, they also allow us to investigate how an equation describing some physical property behaves for small (or large) values for one of the independent variables. 

Special care has to be taken in the representations of angles of 180°, which are wrongly represented as a cusp. To correct this problem with the slope going to zero, trigonometric functions as exemplified in Eq. (5) can be applied [13-15], Close to a maximum this correction may lead to convergence problems, but this price is worth paying in most cases. 

Since the eigenfunctions are unit vectors, an arbitrary function is representable as an eigenfunction expansion. This is because, based on Fourier s theorem that an arbitrary function can be expanded by the series expansion of trigonometrical functions, a function is always expanded by the eigenfunctions of the translational motions, which are trigonometrical functions. 

This representation can be simplified through the use of trigonometric identities to yield... 

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