Trigonometrical series

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

Series for the Trigonometric Functions. In the following formulas, all angles must be expressed in radians. If D = the number of degrees in the angle, and x = its radian measure, then x = 0.017453Z7. 

Correspondingly, the parameter 1/8q tends to zero and we can expand the inverse trigonometric function tan 1/8q in the power series... 

A standard way to represent trigonometric functions is in terms of infinite 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. 

Alternatively, the solution can be worked out through a series of steps similar to those taken for the non-radioactive case. We assume that a solution can be found as a product of a function fit) and a series of trigonometric functions in x such as... 

Fig. 17. 0=S=0 bond angles Inland S=0 bond lengths (r) in XSO2Y sulfones Circles experimental values (for sources see Table 5 and Ref. The empirical relationship 31 = -147.7r + 331.7 was found by a least squares procedure the trigonometrical expression utilizes the remarkable phenomenon of the O O distance being constant (2.484 A) in a large series of sulfone molecules...

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 ... 

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. ... 

The student of thermodynamics should be able to generate such Taylor series expansions for common algebraic and trigonometric functions. 

The last expression is obtained by reducing the integral to integral trigonometric functions and to the asymptotic expansions in a series of these. We restrict ourselves here to the first non-zero term. 

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. 

Since the exponential function may be defined everywhere in the complex plane, we may expand exp(i0) and, using the series expansions for the trigonometric functions, obtain Euler s formula... 

One way to achieve a harmonic series is to set 0 = b. Then the trigonometric identity becomes... 

That T is a series of Bessel rather than trigonometric functions is merely a consequence of using cylindrical polar coordinates (r, j, cz ) for atoms in real space and (R, iji, i/a for points in reciprocal space. Not only is this a convenient framework for describing a helical molecule, but it can lead to economies in computing T. For helices, only Bessel terms with... 

The Taylor series expansion in Chapter 2 makes it possible to derive a remarkable relationship between exponentials and trigonometric functions, first found by Euler ... 

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... 

The series in Eq. (76) converges more rapidly than that in Eq. (68) owing to the presence of trigonometric functions of time in Fn(t). For the case of a simple cubic lattice, we obtain12... 

Thus far, most f.f.t. computations for n.m.r. have been performed sequentially by general-purpose computers that typically require from 10-200 sec to compute a 4,096-point transform. The values of the trigonometric functions are usually obtained from a look-up table stored in the computer alternatively, they may be calculated directly by use of about seven terms of a suitable, infinite series. 

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