The hyperbolic functions

This is called the circular measure of an angle and, for this reason, trigonometrical functions are sometimes called circular functions. This property is possessed by no plane curve other than the circle. For instance, the hyperbola, though symmetrically placed with respect to its centre, is not at all points equidistant from it. The same thing is true of the ellipse. The parabola has no centre. 

If Z denotes the length of the arc of any hyperbola which cuts the a-axis at a distance r from the centre, the ratio 

But this relation is practically that developed for cos x, on page 286, uc, of course, being written for u. The ratio xja is 

The remaining four hyperbolic functions, analogous to the remaining four trigonometrical functions, are tanh u, cosech u, sech u and coth u. Values for each of these functions may be deduced from their relations with sinh u and cosh u. Thus, 

Unlike the circular functions, the ratios x/a, yja, when referred to the hyperbola, do not represent angles. An hyperbolic function expresses a certain relation between the coordinates of a given portion on the arc of a rectangular hyperbola. 

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The transducer function defines the efficiency of the system to translate receptor stimulus into response and defines the efficacy of the agonist. Specifically, it is the fitting parameter of the hyperbolic function linking receptor... 

The first of these functions is effectively the sum of two sin expatta a shown in Fig. 14a, while the hyperbolic sine (sink) is the Btehce . (W) and Rg. 14b]. It should be noted that the hyperbolic functions have do period. They are periodic in the imaginary argument 2nr/. 

Thus, the various relations between the hyperbolic functions can be derived as carried out above for the circular functions. For example,... 

One can also state [02] in terms of pressure described by the hyperbolic function of equation 4.9 giving the noncooperative, hyperbolic Mb curve in Figure 4.9 ... 

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... 

A linear reciprocal transformation of a function of the form [f(a) = a/(l + a)], such that l/f(a) is plotted on the vertical axis and 1/a is plotted on the horizontal axis. In the case of one-substrate enzyme kinetics, the hyperbolic function is ... 

This method is widely used because it provides hnear transformation of the hyperbolic function describing the rate saturation process. Double-reciprocal plots can be reasonably accurate if rate data can be obtained over a reasonable range of saturation, say from 0.3 E ax to 0.8 E ax. 

The function fct d) has no physical basis. According to Heiden (6), it should fulfill only two conditions it should be smooth and continuous and have finite values for d < dcut oS. The value of dclH 0[f should be larger than the van der Waals radius of any atom in the molecule under consideration. In the program LipoDyn, the Ghose and Crippen parameters (18) as well as the Broto and Moreau parameter set (16) are implemented, with the the exponential function used by Fauchere (cxp(-d)) (3), the hyperbolic function defined by Audry 1/(1 + d) (2) and the parameterized Fermi distance function used by Testa and coworkers (1). 

In practice, the values of coefficients A, B and C can be determined by conducting several experiments for the same solute at different flow rates. The method of multiple linear regression is then used to find the hyperbolic function that best matches the experimental values. 

Figure 2.18 Plots of the hyperbolic functions (a) y=sinh xand (b) y= cosh x compared with the exponential functions y= e, y=je x and y= - e x...

The hyperbolic functions are certain combinations of exponentials ande. ... 

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. 

It is important to note that when the slab is very thin (h -> 0) the hyperbolic functions can be replaced by their arguments, and Eq. (16.272) adopts the form... 

In all the formulas above, the terms of the order of k2 were neglected, as well as the terms proportional to e (except, of course, the arguments of the hyperbolic functions). Evidently, (1> (p p. Hereafter we confine our discussion to the most interesting long-time limit case, when 1. Then all the terms proportional to k can be neglected, so we can write... 

We can begin with a general solution of (12-289a). This can be written either as exponentials or in terms of the hyperbolic functions sinh and cosh. The latter turns out to be more convenient. Hence,... 

For x -> 0 or Pm 0 (which implies Tu -> 0), all of the hyperbolic functions approach their arguments, and Eq. (10.64) reduces to the temperature distribution in the gas resulting from conduction alone,... 

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 ... 

I. Conversion Formula.—Corresponding with the trigonometrical formulae there are a great number of relations among the hyperbolic functions, such as (5) above, also... 

IV. Integration of the hyperbolic functions.—A standard collection of results of the differentiation and integration of hyperbolic functions, is set forth in the following table —... 

Numerical values of the hyperbolic functions may be computed by means of the series formulae. 

Notice some expressions for the hyperbolic functions sinh(x) = exp exp(—-v)... 

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