Inverse sine function

Table 2.1 Some Useful Relationships Involving the Hyperbolic Sine and Inverse Hyperbolic Sine Function...

Introduction of the inverse hyperbolic sine function encourages us to take Eq. (2.24) a bit further and derive an expression for 17 itself. Before continuing, let us remember the following ... 

Fig. 1. Top Scheme of an inversion recovery experiment 5rielding the longitudinal relaxation time (inversion is achieved by mean of the (re) radiofrequency (rf) pulse, schematized by a filled vertical rectangle). Free induction decays (fid represented by a damped sine function) resulting from the (x/2) read pulse are subjected to a Fourier transform and lead to a series of spectra corresponding to the different t values (evolution period). Spectra are generally displayed with a shift between two consecutive values of t. The analysis of the amplitude evaluation of each peak from — Mq to Mq provides an accurate evaluation of T. Bottom the example concerns carbon-13 Tl of irans-crotonaldehyde with the following values (from left to right) 20.5 s, 19.8 s, 23.3 s, and 19.3 s.

This can be read as x is the angle whose sine is y. The arcsine function is also called the inverse sine function, and another notation is also common ... 

The FT of step function is a sine function of width inversely proportional to (see Fig. 4) ... 

Thus, finite acquisition time causes a convolution of NMR spectrum with sine function. This manifests itself in peak broadening and presence of sine wiggles . The broadness of the NMR peak is thus dependent not only on relaxation rate but also on the maximum evolution time. Both effects correspond to the Fourier Uncertainty Principle [53] stating that, in general, the broadness of time representation and frequency representation are inversely proportional to each other. 

Taking the inverse of the sine function of both sides, we find that... 

Similarly to the magnetic analyzer, ion separation is based on the circular motion of a charged species in a magnetic field in ICR instruments as well. The difference is that in the ICR, ions undergo several full cycles in the ICR cell. Other differences, such as the application of electrostatic trapping fields, are also crucial and the ion motion is, in fact, much more complex than implied by the simplified discussion below. For the technically inclined reader, we recommend the book by Marshall and Verdun [42]. If only ions with the same mtz ratios are present, such as indicated in Fig. 20, ion motion can be related to a regular sine function, the frequency of which (co) is inversely proportional to the mtz ratio ... 

The energy of a de Broglie wave is inversely proportional to the square of its wavelength. The wavelength of the wave function is the value of x such that the argument of the sine function in Eq. (15.3-10) equals 2ti. 

The total tunneling ray power excited on the endface, P,r(0), is found by integrating Eq. (8-3) over the range 0 Hj]. This is facilitated by integrating the second term within the curly brackets by parts in order to remove the inverse sine function, and then amalgamating with the integral of the third term. The substitution = M ., (1 — reduces the combined integral to the form of Eq. (37-116). If we normalize with the total bound-ray power of Eq. (4-16) then... 

Brigham (1974) shows that the inverse FT of the sine function returns our rectangular pulse - see the derivation that leads up to his Equation 2.13. ... 

Inverse Hyperbolic Functions If x = sinh y, then y is the inverse hyperbolic sine of x written y = sinh" x or arcsinh x. sinh" x = log x + + 1)... 

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. 

One should note that the phase shift becomes time-independent and maximal for a = 1, i.e., at the resonance condition v = vG. The frequency spectrum 4>(a) bears a sine shape with a bandwidth inversely proportional to the number of oscillations of the gradient field (Fig. 4). Such a behaviour was also predicted in Ref. 15. Recording in a systematic way the phase shift as a function of vG without space encoding would be a very fast and efficient method to scan in a whole object the possible frequencies of spin motions. 

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. 

The one exception in which phase contrast is not due to the dissipation arises when the tip jumps between attraction phases (>90°) and repulsion phases (<90°). Since sine is a symmetric function about 90°, the phase changes symmetric even if there are no losses in the tip-sample interaction. The relative contribution of the repulsive and attractive forces can be estimated experimentally from the frequency-sweep curves in Fig. lib by measuring the effective quality factor as Qe=co0/Ao)1/2, where Ago1/2 is the half-width of the amplitude curve. The relative contribution of the attractive forces was shown to increase with increasing the set-point ratio rsp=As/Af. Eventually, this may lead to the inversion of the phase contrast when the overall force becomes attractive [110,112]. The effect of the attractive forces becomes especially prominent for dull tips due to the larger contact area [147]. 

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