Periodic functions, Fourier expansions

Each logarithm in the last temi can now be expanded and the (—n)th Fourier coefficient arising fi om each logarithm is — jn) zk-Y- To this must be added the n = 0 Fourier coefficient coming from the first, f-independent term and that arising from the expansion of second term as a periodic function, namely. 

Hence we see that this simple periodic function has just two terms in its Fourier series. In terms of the Sine and Cosine expansion, one finds for this same f(t)=Sin3t that an = 0, bn =... 

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]... 

The term Fourier coefficient originates from the theory of Fourier series, in which periodic functions are expanded based on a set of sine- and cosine-functions. The expansion coefficients are called Fourier coefficients. 

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

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... 

The continuum limit of the Hamiltonian representation is obtained as follows. One notes that if the friction function y(t) appearing in the GLE is a periodic function with period T then Eq. 4 is just the cosine Fourier expansion of the friction function. The frequencies coj are integer multiples of the fundamental frequency and the coefficients Cj are the Fourier expansion coefficients. In practice, the friction function y(t) appearing in the GLE is a decaying function. It may be used to construct the periodic function y(t T) = Y(t TiT)0(t-... 

Since the torsion itself is periodic, so too must be the torsional potential energy. As such, it makes sense to model the potential energy function as an expansion of periodic functions, e.g., a Fourier series. In a general form, typical force fields use... 

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.,... 

As the Fourier expansion of the periodic delta function, we obtain... 

For a periodic crystal, we introduce the following Fourier expansion of the linear density-response function... 

Note The designer should not get confused by the fact that the first term in the expansion is sometimes called a0/2, sometimes ao, or sometimes something else altogether. Either way, in any Fourier expansion of any arbitrary periodic function, the first term is always the area under the waveform calculated over one time period (i.e. its arithmetic average). 

Since both bulk and surface states are molecular in character, the wave functions of atoms in both types of position can be calculated by the same method. Appelbaum and Hamann [70] assume two-dimensional periodicity along the surface and make the same Fourier expansion of the pseudo-wave function as for the bulk, except that at each of a set of discrete surface normal co-ordinates a different set of expansion coefficients is used. These sets can be integrated from outside the surface into the bulk. Well inside the bulk, these wave functions are matched to bulk states of similar lateral symmetry and the matching condition determines energies and wave functions. 

Fourier s theorem determines the law for the expansion of any arbitrary function in terms of sines or cosines of multiples of the independent variable, x. If f(x) is a periodic function with respect to time, space, temperature, or potential, Fourier s theorem states that... 

In the third step, an elementary solution of the system in appropriate mathematical form is chosen for the initial disturbance. Typically the complex form of the Fourier representation of periodic functions, although the more cumbersome form of an expansion in a series of sine and cosine termis may equally well be used. For example, the elementary solution might be chosen to be the normal mode... 

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