Arctangent function

Therefore, if the total phase shift is known as a function of the energy, the positions of the resonances can be determined by fitting it with a linear combination of arctangent functions plus a smooth polynomial background. See, for example, Ref. [74] and references therein. 

MRS spectra from the human brain. In order to improve the robustness of the X2 minimization, an internal parameter R was used, defined by r = exp(— R ). The arctangent function, however, is preferable as it has a continuous derivative around R = 0. The equal asymptotic behaviour of this function towards 0 and 1 ensures that r remains within this valid range. We have subsequently found that using r = (arctan(7 ) + Tr/2)/n gives better performance. The fit can be conveniently performed using only the real part of the complex spectrum. Thus the normalized x2 is given by... 

Lorentzian and arctangent functions. The separate components of this function are also shown. 

Using the arctangent function, the angle of the external magnetic field can be calculated as ... 

Sketch graphs of the arcsine function, the arccosine function, and the arctangent function. Include only the principal values. g]... 

Just as with a transformation from cartesian coordinates to polar coordinates, we do not necessarily use the principal value of the arctangent function, but must obtain an angle in the proper quadrant, with 0 ranging from 0 to 27t. 

Some forms of BASIC may not have an arccosine (ACOS) function, even though they all have an arctangent function, ATAN or ATN. Perhaps this omission has come about because arccosine (and arcsine) can be expressed in terms of arctan. In fact, some BASIC manuals give the explicit formulas for computing arccosine and arcsine. 

Thus, since the relaxation spectrum I(t) contains a(T) as shown in Eq. (8.3) and a(T) can be determined with E as a parameter, I(x) can also be determined experimentally by use of Eq. (8.3) and parameter E from the data of both ESR and NMR observations, data of line widths, or second moments. Therefore, if both values of I(x) obtained from ESR and NMR data are identical with a certain value of E, this value E must be the true activation energy of the molecular motion, provided that ESR and NMR observe the same molecular motions. However, further evidence concerning this problem is needed. Due to the nature of I(x), the symmetric function, and characteristic of the arctangent function appearing in Eq. (8.1) and Eq. (8.2), the following relation can be obtained ... 

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