Trigonometric Fourier series

An almost periodic function is uniquely defined in average by a trigonometric Fourier series... 

Analytical solution is possible only when the reaction in the body of the reactor is first or zero order, otherwise a numerical solution will be required by finite differences, method of lines or finite elements. The analytical solution proceeds by separation of variables whereby the PDE is converted into ODEs whose solutions are in terms of trigonometric functions. Satisfying all of the boundary condtions makes the solution of the PDE an infinite series whose development is quite elaborate and should be sought in books on Fourier series or partial differential equations. 

Notice that we have approximated a discontinuous function by a continuous one. It turns out that any function in L —1, 1] can be approximated by trigonometric polynomials — this is one of the important results of the theory of Fourier series. ... 

In words, the difference Patterson function is a Fourier series of simple sine and cosine terms. (Remember that the exponential term is shorthand for these trigonometric functions.) Each term in the series is derived from one reflection hkl in both the native and derivative data sets, and the amplitude of each term is (IFHpI — IFpl)2, which is the amplitude contribution of the heavy atom to structure factor FHp. Each term has three frequencies h in the u-direction, k in the v-direction, and l in the w-direction. Phases of the structure factors are not included at this point, they are unknown. 

As a second example, suppose that the crystal has three mutually perpendicular, two-fold axes of symmetry. In this case, if we take these axes as the axes of coordinates, each term in the Fourier series may contain the product of three cosines, but, unless the coordinate planes are also planes of symmetry, the series must contain terms with trigonometric sines. These sines, however, must occur in pairs. A term cannot contain the product of one sine and two cosines, nor can it be the product of three sines for the term must have the same value when we change the algebraic signs... 

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... 

A Fourier series is a type of infinite trigonometric series by which any kind of periodic function may be expressed. Now the one essential property of a crystal is that its atoms are arranged in space in a periodic fashion. But this means that the density of electrons is also a periodic function of position in the crystal, rising... 

However, the expansion above is still impractical for our purposes, because the functions As(q), Bs(q),. .. still need to be expanded in an infinite Fourier series of the angles e.g., we should write A q) = J2kez akexp(i(k,qj). It is more convenient to work with trigonometric polynomials, so that every part of the expansion contains only a finite number of terms. To this end, we introduce a Fourier cutoff by splitting every function of the angles in an infinite number of slices that contain only a finite number of Fourier modes. This may be done in many arbitrary ways, so let us illustrate just one method. We choose an arbitray integer K, ad write, e.g.,... 

The following table of trigonometric identities should be helpful for developing Fourier series. 

Taylor series expansions, as described above, provide a very general method for representing a large class of mathematical functions. For the special case of periodic functions, a powerful alternative method is expansion in an infinite sum of sines and cosines, known as a trigonometric series or Fourier series. A periodic function is one that repeats in value when its argument is increased by multiples of a constant L, called the period or wavelength. For example. 

An alternative modulation system is shown in Fig. 12.12. This technique uses an electronic switch rather than a carrier and a mixer. To see why this works, consider that switching m t) on and off is the same as multiplying m t) with a square wave s(t) having frequency fp, x t) = m t)s t). Using Fourier series, s (t) can be written as a trigonometric expansion ... 

The techniques of Fourier analysis are perhaps best appreciated initially by an example. Consider the ideal continuous-time square wave of period T s, having a maximum value of A and a minimum value of 0 as shown at the top of Fig. 20.68. Because of the periodic nature of the waveform, the technique of Fourier series (FS) analysis applies, and the resulting decomposition into an infinite sum of simple trigonometric... 

The Fourier series is one of the general class of trigonometric series described by the expression... 

Otherwise we may fall back upon Maclaurin s expansion in ascending powers of a , the constants being positive, negative or zero. This series is particularly useful when the terms converge rapidly. When the results exhibit a periodicity, as in the ebb and flow of tides annual variations of temperature and pressure of the atmosphere cyclic variations in magnetic declination, etc., we refer the results to a trigonometrical series as indicated in the chapter on Fourier s SBries. - ... 

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. 

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