Function cosine

Differential cross-sections for particular final rotational states (f) of a particular vibrational state (v ) are usually smoothened by the moment expansion (M) in cosine functions mentioned in Eq, (38). Rotational state distributions for the final vibrational state v = 0 and 1 are presented in [88]. In each case, with or without GP results are shown. The peak position of the rotational state distribution for v = 0 is slightly left shifted due to the GP effect, on the contrary for v = 1, these peaks are at the same position. But both these figures clearly indicate that the absolute numbers in each case (with or without GP) are different. 

FIGURE 6.1 The energy due to conformation around a single bond represented by a cosine function. 

Most types of motion due to vibration occur in periodic motion. Periodic motion repeats itself at equal time intervals. A typical periodic motion is shown in Figure 5-3. The simplest form of periodic motion is harmonic motion, which can be represented by sine or cosine functions. It is important to remember that harmonic motion is always periodic however, periodic motion is not always harmonic. Harmonic motion of a system can be represented by the following relationship ... 

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. 

The step change is close to the situation where the sensor is suddenly moved from one place to another having a different state of the measured quantity. The exponential change could, for example, be the temperature change of a heating coil or some other first-order system. Finally, the velocity fluctuations of room air can be approximated with a sine or cosine function. 

The PECD measurement clearly takes the form of a cosine function with an amplitude given entirely in terms of the single chiral parameter, b. It therefore provides exactly the same information content as the y asymmetry factor dehned above [Eq. (8)]. Experimental advantages of examining the PECD rather than the single angular distribution /p(0) are likely to include some cancellation of purely instrumental asymmetries (e.g., varying detection efficiency in the forward-backward directions) and consequent improvements in sensitivity. 

The radial frequency co of a periodic function is positive or negative, depending on the direction of the rotation of the unit vector (see Fig. 40.5). co is positive in the counter-clockwise direction and negative in the clockwise direction. From Fig. 40.5a one can see that the amplitudes (A jp) of a sine at a negative frequency, -co, with an amplitude. A, are opposite to the values of a sine function at a positive frequency, co, i.e. = Asin(-cor) = -Asin(co/) = This is a property of an antisymmetric function. A cosine function is a symmetric function because A -Acos(-co/) = Acos(cor) = A. (Fig. 40.5b). Thus, positive as well as negative... 

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 ... 

Fourier coefficients, represent the frequencies of the cosine functions. In the remainder of this chapter we use the shorthand notation F(n) for [A - jB ]. 

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 representation of tp(x, t) by the sine function is completely equivalent to the cosine-function representation the only difference is a shift by A/4 in the value of X when t = 0. Moreover, any linear combination of sine and cosine representations is also an equivalent description of the simple harmonic wave. The most general representation of the harmonic wave is the complex function... 

A wave function composed in this way from the contributions of single atoms is called a Bloch function [in texts on quantum chemistry you will And this function being formulated with exponential functions exp(inka) instead of the cosine functions, since this facilitates the mathematical treatment]. 

The computer evaluates the cosine functions appearing in Eq. (38) from their series expansions, as given by Bq. (1-34). As M is usually a large number, the time required for the evaluation of the sum can be considerable. However, die arguments of the cosines are simply related because the data points are separated by the constant interval Ax. Given the relation... 

Performing the indicated integration then leads to the probability of incident molecules striking the wall coming from angle Q, which is P(Q) dQ, in the form of a simple cosine function,... 

The use of sine or cosine functions in FID data processing is an essential tool in 2D NMR. 

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. 

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