Trigonometric series

A. Zigmund, Trigonometrical Series, Dover, New York, 1955, Chap. III. 

A mathematical procedure for expressing any singlevalued periodic function as the infinite series summation of a succession of sine waveforms of higher order, such that f(x), a function with a period of 2tt, can be treated as a trigonometric series ... 

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... 

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. - ... 

I. The development of a trigonometrical series of sines. Suppose it is required to find the value of... 

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. 

Under these conditions, the last trigonometric series is successively evaluated... 

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

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