Sine and cosine functions

After the forces are evaluated for each cylinder of a multistage compressor, all forces must be summed in the x and y direction. For the max imum shaking forces, the value of the crank angle, which contributes the maximum force, should be used. This involves taking the respective sine and cosine functions to their maximum. For example, a vertical cylinder will have the maximum component force at a crank angle of 0 and 180 . At this time, the horizontal components, primary and secondary, are zero. 

A relationship, known as Euler s formula, exists between a complex number [x + jy] (x is the real part, y is the imaginary part of the complex number (j = P )) and a sine and cosine function. Many authors and textbooks prefer the complex number notation for its compactness and convenience. By substituting the Euler equations cos(r) = d + e -")/2 and sin(r) = (d - e t )l2j in eq. (40.1), a compact complex number notation for the Fourier transform is obtained as follows ... 

As explained before, the FT can be calculated by fitting the signal with all allowed sine and cosine functions. This is a laborious operation as this requires the calculation of two parameters (the amplitude of the sine and cosine function) for each considered frequency. For a discrete signal of 1024 data points, this requires the calculation of 1024 parameters by linear regression and the calculation of the inverse of a 1024 by 1024 matrix. 

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. 

The shorter the wavelength, the faster its decay. Mineral scale heterogeneities in rocks disappear long before meter-scale or even larger heterogeneities. This concept can be extended to any arbitrary combination of periodic functions in Section 2.6, we have already met the idea that any function bounded over an interval can be expanded as a sum of sine and cosine functions. Shorter wavelengths will decay much faster... 

A disadvantage of Fourier compression is that it might not be optimal in cases where the dominant frequency components vary across the spectrum, which is often the case in NIR spectroscopy [40,41], This leads to the wavelet compression [26,27] method, which retains both position and frequency information. In contrast to Fourier compression, where the full spectral profile is fit to sine and cosine functions, wavelet compression involves variable-localized fitting of basis functions to various intervals of the spectrum. The... 

The asterisk designates the complex conjugate. Moreover, we note that the above Eqs. 2.46 and 2.47 imply positive as well as negative frequencies. In some physics applications, an appearance of negative frequencies may be confusing only positive frequencies may have physical meaning. In such cases one may rewrite the above inverse tranform in terms of positive frequencies, using a well-known relationship between the complex exponential function and the sine and cosine functions. 

Functions having a property f(x a) =f x) are known as periodic functions with a period a, and are said to be many-to-one functions. In the examples given above, the period for the sine and cosine functions is 2n, while that for the tangent function is n. 

We can see from Table 2.5 and Figure 2.17 that the sine and cosine functions both have as domain the set of real numbers. The domains of the tangent and reciprocal trigonometric functions are different, however,... 

The hyperbolic sine and cosine functions sinh - and cosh. v are defined in terms of the sum and difference of the exponential functions e and e. respectively ... 

As r — oo the wavefunctions are oscillatory sine and cosine functions, as shown by Eqs. (2.14). For small r the wavefunctions of the continuum are functionally identical to the bound wavefunctions, differing only in their normalization. Since continuum waves extend to r = oo they cannot be normalized in the same way as a bound state wavefunction. We shall normalize the continuum waves per unit... 

Although the properties of the/and g functions are outlined in chapter 2, it is worth summarizing their properties here.8 The / and g coulomb functions are termed regular and irregular since asr— 0,/< rt+ and g oc r (. Due to the r = 0 behavior of the g function, in H only the/wave exists. As r — for Wt > 0 the/ and g waves are sine and cosine functions, and if Wt > 0, jiv simply specifies the phase of the wavefunction relative to the hydrogenic/wave. If Wt < 0 the/and g waves both have exponentially increasing and decreasing parts, and, as we have seen in Chapter 2, only if... 

Equation 3.48 is of course the same equation as we have solved before, e.g. for the particle in a box. Its solutions are simple sine and cosine functions of angular variable, which repeats itself every 2n radians. The boundary conditions for the wavefunction are therefore different from those for the particle in a box. There is no requirement that iff must be zero anywhere instead, it must be single valued, which means for any 0,... 

In Fourier compression, each profile (x ) is essentially decomposed into a linear combination of sine and cosine functions of different frequency. If the spectrum x is considered to be a continuous function of the variable number m, then this decomposition can be expressed as ... 

The basis set using a finite basis set necessarily leads to an inexact wavefunc-tion, in much the same way that representing a function by a finite Fourier series of sine and cosine functions necessarily gives an approximation (albeit perhaps an excellent one) to the function. 

FIGURE 1.1 Definition of the sine and cosine function, in terms of positions on a circle with radius 1. 

Taylor series for the sine and cosine function are also often useful ... 

The algorithm for the FFT (the reverse butterfly in our case) is well known (ref. 1,2) and will not be discussed here in detail. On the other hand, the FHT has been often neglected in spite of some advantages it offers. Due to the fact that both transformations rotate the time domain into the frequency space and vice versa, the only conceptual difference between both transformations is the choice of basis vectors (sine and cosine functions vs. Walsh or box functions). In general, the rotation or transformation without a translation can be written in the following form (ref. 3) ... 

The time dependence of any periodic function can always be expressed in terms of sine and cosine functions in this sense, it is very convenient to apply the complex number representation using the following equation [135] ... 

We can also use the sine and cosine functions to describe any general rotation, not confined to the four points of the compass ... 

In each case, r times the multiplier in front of the Hamiltonian becomes the argument (in radians) of the sine and cosine functions when we write out the time course of the spin state. 

The first term in [ ] is identically zero because, as just noted, l)( o> oo) = 1. The exponential in the remaining integral can be expanded into sine and cosine functions,... 

We have indicated that interference and reinforcement effects depend both on the positions of atoms in a structure and the number of electrons associated with each atom. A quantitative treatment of these effects makes use of the important structure-factor equation which represents the addition of waves (sine and cosine functions) from each atom within a unit cell. All waves are of the same lengths but amplitudes and phases may differ. The structure-factor equation, to be given here but not derived, deals with the relative intensities of the reflected rays rather than with the absolute amplitudes or intensities of reflected rays (which depend on the amplitudes and intensities of the x-rays used as a source). The relative intensity, /, of a ray of indices hkl, from a set of planes hkl, is... 

Analysis — Any arbitrary time-dependent function may be synthesized by adding together sine and cosine functions of different frequencies and amplitudes, a process known as synthesis [i]. Conversely, the determination of the amplitudes and frequencies of the sine and cosine waves that make up a time-dependent signal (or noise), for example v(f) or I(t), is known as analysis (or decomposition). Thus, for a signal defined over some time period T, analysis results in the determination of the amplitudes a and b , as a function of frequency in the expression... 

This suggests, as sine and cosine functions are trivially related, that perhaps one can define some sine and cosine SETO forms with angular parts written as follows ... 

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