Hyperbolic functions expansion

Equation 26 was derived directly and contains no mathematical approximations. A similar equation was derived by Healy et al. (30). If the hyperbolic functions are approximated by a linear expansion, Equation 26 can be approximated... 

Mathematically this happens because the first nonlinear term in the Taylor expansion series of exp x) is x 2 while for the hyperbolic function sinh(x) it is X /6. 

In the experiments we always deal with film having thickness much less than the wavelength of the observable surface waves (see Section III.A). This means that in the experiments the long-wavelength limit (Iwl) is observed. Since the Iwl corresponds with the conditions Kh<. and w/i bending mode can be found from (5.24) and (5.25) by series expansion of the hyperbolic functions. The result for the squeezing mode is ... 

In the limit ajh -> 0, a straightforward expansion of the hyperbolic functions for large values of a yields... 

This accords with the original result of Lorentz (LI 1), obtained by employing a first reflection. In the opposite case, where the sphere is very near the wall, i.e., ajH- oo (where H = h — a), one finds from (131) that a. - 2Hla. Upon expansion of all the hyperbolic functions in (130) for small a, one obtains... 

The series expansions for the hyperbolic functions are similar to Eqs. (4.55) and (4.56), except that all terms have plus signs ... 

The analytical terms are computed through reducing them as such to contain the hyperbolic functions and then appl5ang the approximations of type (2.45). With this recipe we firstly evaluate for the fluctuation width the expansion of the term (Putz, 2009) ... 

The integrals I0 and /2 arise from a Fourier expansion for hyperbolic cosines of trigonometric argument [45]. With this potential, Parker calculated both the average number of collisions to establish rotational equilibrium, Zf, and that required to establish vibrational equilibrium, Z , as functions of temperature. One finds... 

If we carefully observe the expression of the characteristic function of the residence time distribution for the evolution of a liquid element (q>(s,Hd), relation (4.201)), we can notice that it is difficult to compute the expressions of the derivatives cp (0, Hd) and (p"(0, H ). Using the expansion of the hyperbolic sine and cosine respectively as multiplication series, we obtain the following simplified expression for the characteristic function ... 

During an experiment the sample is cold, ca 20K (= 14 cm ). The lowest internal vibrations are typically about 300 cm and the hyperbolic sine function will, except at the very lowest energies, have an argument greater than ten. The argument of the Bessel function is, therefore, less than 10 and it can be safely represented by the first term of its power series expansion. Where, for an arbitrary argument x ... 

Hyperbolic and Inverse Hyperbolic Table of expansion of certain functions into power series... 

Results in Eq. 8.114 are identical to those derived by Markham [68] for an analogous Hamiltonian worked out to describe spectra of electron traps in crystals. In the classical limit, jS ti(Ok, we can take a first-order expansion of the hyperbolic cotangent function coth (phcbk/l) IksT/fimk, thus obtaining the expressions... 

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