Sine function plots

Figure 2—cont d The final residual curve coincided well with the crystalline Gaussian/sine function (C). The sum of these resolved profiles was compared with observed plots (D). 

Fig. 9.7. Example of a Fourier transformation. The signal in time domain at left is the sine function, the magnitude of its Fourier transform is plotted at right. It is noted that F(co) is band limited.

The boundary condition relevant to motion in a circle is different from that required for a particle in a box, where the wavefunction had to go to zero at the ends of the box. For circular motion the wavefunction has to match up with itself after one complete revolution of the circle. This requires the circumference of the circle to be equal to a whole number of wavelengths. The situation where five wavelengths fit into the circle is illustrated for the sine function in Figure 5.3a. The plot for the cosine function would be similar, but rotated through 90 . If this condition is not met the waves will not coincide with one another after one complete revolution, and multiple values of will be obtained for any particular point on the circle, as shown in Figure 5.3b. As we saw in Section 1.4.5,... 

Like the sine function, the more complicated wave functions for atomic orbitals can also have phases. Consider, for example, the representations of the Is orbital in T Figure 9.38. Note that here we plot this orbital a bit differently from what is shown in Section 6.6. The origin is the point where the nucleus resides, and the wave function for the Is orbital extends from the origin out into space. The plot shows... 

What do these wavefunctions look like Figure 10.6 shows plots of the first few wavefunctions. All of them go to zero at the sides of the box, as required by the boundary conditions. All of them look like simple sine functions (which is what they are) with positive and negative values. 

Figure 1.17 shows the MATLAB-generated figure that contains two subplots. The first subplot command picks the first row of this panel and plots the cosine function in it. The second picks the second row and plots the sine function in it. In this way, the two subplots are vertically aligned. 

Figure 4.2 Plot of the sine function fix) at its Hermite interpolating polynomial p x), fitting the function and first derivative values at each end point.

Figure 3.10 Effect of different window functions (apodization functions) on the appearance of COSY plot (magnitude mode), (a) Sine-bell squared and (b) sine-bell. The spectrum is a portion of an unsymmetrized matrix of a H-COSY I.R experiment (400 MHz in CDCl, at 303 K) of vasicinone. (c) Shifted sine-bell squared with r/4. (d) Shifted sine-bell squared with w/8. (a) and (b) are virtually identical in the case of delayed COSY, whereas sine-bell squared multiplication gives noticeably better suppression of the stronger dispersion-mode components observed when no delay is used. A difference in the effective resolution in the two axes is apparent, with Fi having better resolution than F. The spectrum in (c) has a significant amount of dispersion-mode line shape.

Fig. 7-1. The annual average insolation and average zenith angle as a function of the sine of latitude. The zenith angle has been multiplied by a factor of 10 so that its variation can be seen. The sine of latitude is used as the ordinate in all of these plots because it reflects the relative surface area at each latitude.

Fig. 7-4. Specified distributions of land fraction, land cloud cover, and ocean cloud cover as functions of the sine of latitude. Also plotted is the albedo calculated for these values of land and cloud and for the temperatures of Figure 7-6. Negative values of SIN(LATITUDE) correspond to the Southern Hemisphere.

Fig. 7-6. Calculated values of the annual average temperature as a function of the sine of latitude, plotted as a solid line, compared with observed values, plotted as squares.

The other plots are made with the software TABLECURVE. The special function F2 used there is a log-normal relation and F3 is a sine-wave function. Usually a ratio of low degree polynomials also provides a good fit to bell-shaped curves here five constants are needed. The Gamma distribution needs only one constant, but the fit is not as good as some of the other curves. The peak, especially, is missed. 

In practice, the phase shift and the modulation ratio M are measured as a function of co. Curve fitting of the relevant plots (Figure 6.6) is performed using the theoretical expressions of the sine and cosine Fourier transforms of the b-pulse response and Eqs (6.23) and (6.24). In contrast to pulse Jluorometry, no deconvolution is required. 

Working independently, A.Abakonovicz in 1878 and C.V. Boys in 1882 devised the integraph, an instrument that drew the integral of an arbitrary function when the latter was plotted on a suitable scale on paper. A device for finding trigonometric functions (sines and cosines), known as harmonic analyzer was devised in 1876 by Lord Kelvin. 

Figure 2.17 Plots of the trigonometric functions sine (dot-dash line), cos 0 (full line), and tane (dashed line) for -2n s, 0 < 2n. The principal branch of each function is shown by the thick lines. The dotted vertical lines at odd multiples of n 2 indicate the points of discontinuity in the tangent function at these values of e...

The raw data or FID is a series of intensity values collected as a function of time time-domain data. A single proton signal, for example, would give a simple sine wave in time with a particular frequency corresponding to the chemical shift of that proton. This signal dies out gradually as the protons recover from the pulse and relax. To convert this time-domain data into a spectrum, we perform a mathematical calculation called the Fourier transform (FT), which essentially looks at the sine wave and analyzes it to determine the frequency. This frequency then appears as a peak in the spectrum, which is a plot in frequency domain of the same data (Fig. 3.27). If there are many different types of protons with different chemical shifts, the FID will be a complex sum of a number of decaying sine waves with different frequencies and amplitudes. The FT extracts the information about each of the frequencies ... 

Below is the symmetrized, magnitude, COSY-60 spectrum of camphor with the one-dimensional spectrum plotted to the same scale. This spectrum was collected using the full phase cycling normally applied to suppress artefacts. Each FID consisted of IK data points, and 256 increments were used. The data were multiplied by a sine bell window function in both dimensions to improve the appearance of the contour plot. 

Figure 3. TOCSY and NOESY contour plots are shown for Ppep-4 for the amide/aromatic region in Hj0(90%)/Dj0(10%) (left) and for the aH region in DjO (right). Ppep-4 peptide concentration was 20 mg/mL in 20 mM potassium phosphate, pH 6.3, 40°C. The NOESY mixing time was 0.2 s, and the TOCSY spin iock time was 40 ms. The data were zero-fiiied to 1024 in t1. The raw data were multiplied by a 30° shifted sine-squared function in t1 and t2 prior to Fourier transformation.

A contour plot is shown in Fig. 7.8. Note that this function is cylindrically-symmetrical about the z-axis with a node in the x, y-plane. The eigenfunctions 21 1 are complex and not as easy to represent graphically. Their angular dependence is that of the spherical harmonics 7i i, shown in Fig. 6.4. As deduced in Section 4.2, any linear combination of degenerate eigenfunctions is an equally-valid alternative eigenfunction. Making use of the Euler formulas for sine and cosine,... 

Fig. 4. High-frequency shifted portions of the 500 MHz H NMR spectra of sperm whale metMbiCN ) at pH 9.2, 30 "C. (A) In 90 10 = H2O H2O. (B) In 49 51 = H2O H2O. The expanded plots show the three resolved heme methyl signals after resolution enhancement by sine-bell function the vertical scale is arbitrary, frequency scale magnified by a factor of 5. Asymmetry in the expanded peaks is due to overlapping lines of different width. (From ref. 79, 1987, with permission from the publishers.)... 

The basic sine bell is just the first part of a sin 9 for 9 = 0 to 6 = tv, this is illustrated in the top left-hand plot of Fig. 4.13. In this form the function will give resolution enhancement rather like the combination of a rising exponential and a Gaussian function (compare Fig. 4.11 (j)). The weighting function is chosen so that the sine bell fits exactly across the acquisition time mathematically the required function is ... 

Figure 7.3 The time dependence of the total concentration as a function of the rescaled time Dai, at several values of Da for the autocatalytic dynamics in the closed sine-flow of Eq. (2.66) Da increases from top to bottom, dashed line is the homogeneous result, which is approached as Da —> 0. The inset plots the evolution in terms of the unsealed time, showing that at any given time the amount of C is larger for larger Da, which corresponds to the most inhomogeneous configuration. Dashed line in the inset is the exponential growth of the length of a material line.

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