Trigonometric equation

W. Gautschi. Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math., 3 381-397, 1961. 

The frequency-dependent coefficients in this equation are given separate names and symbols to facilitate discussion. Remember it is these coefficients that determine the behavior of the system the trigonometric functions merely describe the oscillations. The following can be said of the coefficient of the cosine term ... 

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

Tsai and Pagano [2-7] ingeniously recast the stiffness transformation equations to enable ready understanding of the consequences of rotating a lamina in a laminate. By use of various trigonometric identities between sin and cos to powers and sin and cos of multiples of the angle, the transformed reduced stiffnesses. Equation (2.85), can be written as... 

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. 

Legitimate operations on equations include addition of any quantity to both sides, multiplication by any quantity of both sides (unless this would result in division by zero), raising both sides to any positive power (if is used for even roots) and taking the logarithm or the trigonometric functions of both sides. 

Thus the original differential equation (6-90) of the second order has been replaced by the system (6-96) of two first order differential equations in terms of the amplitude a and the phase 9. Moreover, as Eqs. (6-96) contain the small factor (i on the right-hand side, the quantities, a and 9 are small, that is, both a and 9 are slowly varying functions of time and one can assume that during one period T = 2nfca, the trigonometric functions vary but slightly. 

The solutions to equation (2.34) are functions that are proportional to their second derivatives, namely sin(27rv/A) and cos(2jrjc/A). The functions exp[27riv/A] and exp[—27riv/A], which as equation (A.31) shows are equivalent to the trigonometric functions, are also solutions, but are more difficult to use for this system. Thus, the general solution to equation (2.34) is... 

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. 

Since the two members in the last equation cannot be functions of the independent variables x and t, they must be equal to a same constant, which suggests using an exponential form for/(f) and a trigonometric form for g(x). The diffusion equation is indeed identically verified for... 

Because of the periodic properties of the trigonometric functions we know that the integral on the right of equation... 

The rapidly changing trigonometric denominator gives rise to divergences that correspond to the molecular resonances [16,27,28]. We can see that the DOS contains the same divergent term as the transmission, therefore, at least close to the molecular resonances, the transmission seems to be proportional to the scaled DOS, as suggested by the equation defining t (equation (32)). The spectral function (see equation (37)) contains the term Ti + U2, which refers to the contacts... 

These equations satisfy the Fourier analysis for Cartesian coordinates, and the solution can be directly written down in terms of trigonometric functions as follows ... 

We do not discount the possibihty that a two term, four parameter equation could be found using trigonometric or other functions which can produce an energy term that becomes large for soft-soft or hard-hard combinations, but not for others. Clearly, none has been reported to date. 

The solutions of a diffusion equation under the transient case (non-steady state) are often some special functions. The values of these functions, much like the exponential function or the trigonometric functions, cannot be calculated simply with a piece of paper and a pencil, not even with a calculator, but have to be calculated with a simple computer program (such as a spreadsheet program, but see later comments for practical help). Nevertheless, the values of these functions have been tabulated, and are now easily available with a spreadsheet program. The properties of these functions have been studied in great detail, again much like the exponential function and the trigonometric functions. One such function encountered often in one-dimensional diffusion problems is the error function, erf(z). The error function erf(z) is defined by... 

Cosine trigonometric functions, in other words, are given by the real part of the function e10. This means that Equations (80) and (83) may be written... 

Equation 7.A2.6 can be put into a more useful form by using the trigonometric identity ... 

Here R is the distance of the detector from the specimen, the trigonometric factor in Equation (2.2) accounts for the effects of polarisation and the other factors have their usual physical meanings. The amplitude... 

Interpolation consists of finding the correlation between the known points according to the selected basis functions. Hence, we need to search for appropriate equations that fit the behavior of our function / (x). For example, in linear interpolation, the chosen function is a straight line. The most commonly used functional forms are polynomials, rational functions, trigonometric functions and radial functions [10, 19, 21]. 

Another important class of functions encountered in chemistry and physics is the trigonometric functions. Consider the equation x2 + y2 = 1. The set of all points in a plane that satisfy this equation is a circle with radius 1 (Figure 1.1). Any position on the circle could be labeled by the length 9 of the arc which stretches counterclockwise from the positive x-axis to that point. Since the circle has circumference 2jt, only values of 6 between 0 and 2tt are needed to describe the whole circle. 

In order to systematize in a logical form the lattices that are compatible with a periodicity condition, the French physicist Auguste Bravais, in 1845, demonstrated that the lattice points in three dimensions, congruent with the periodicity requirement, are the roots of the following trigonometric equation [2] ... 

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