Other hyperbolic functions

Process engineering and design using Visual Basic 

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The other hyperbolic functions, tanh, cosech, sech andcoth, defined in terms sinh and cosh, are the hyperbolic analogues of the functions tan, cosec, sec and cotan, and are defined as follows ... 

On the other hand if x/m < h2/4m2, the eqffetion for z(t) is of the form of Eq. (25) and the solutions are in terms of expbhential functions of retil arguments or hyperbolic functions. In this case jr(r) is not oscillatory and will simply decrease exponentially with time. 

Functions having the property f -x) =,f x) are called symmetric, or even, functions, whilst those having the property f(-x) — -f(x) are called antisymmetric or odd functions. In our discussion of trigonometric and hyperbolic functions, we have encountered a number of examples of functions that fall into one or other of these categories, as well as some that fall into neither. Symmetric and antisymmetric functions are so called because they are symmetric or antisymmetric with respect to reflection in the y-axis. A close look at Figure 2.17 shows that, since cos0=cos(-0), and sin0=-sin(-0), the cos and sin functions are symmetric and antisymmetric, respectively. Likewise, we can classify the cosh and sinh functions... 

It is evident that Equation (4.11) is of a very general mathematical form (i.e. a hyperbolic function). At low 6 it reduces to Henry s law at high surface coverage, a plateau is reached as 6—>1. Other equations of the same mathematical form as Equation (4.11) have been derived from a classical thermodynamic standpoint (Brunauer, 1945) and by application of the principles of statistical mechanics (Fowler, 1935). 

Other sigmoidal functions, such as the hyperbolic tangent function, are also commonly used. Finally, Radial Basis Function neural networks, to be described later, use a symmetric function, typically a Gaussian function. 

For large xa it reduces (as )3.5.56) does) to the flat layer limit. On the other hand, for low xa 1) and low potentials (both hyperbolic functions replaced by their arguments). (3.5.64) reduces to the corresponding equation for spheres with twice the radius, )3.5.51). For a discussion of this and other approximate... 

Equation 3-56 is a linear, homogeneous, second-order differential equation with constant coefficients. A fundamental theory of differential equations states (hat such an equation has two linearly independent solution functions, and its general solution is the linear combination of those two solution functions. A careful examination of the differential equation reveals that subtracting a constant multiple of the solution function 0 from its second derivative yields zero. Thus we conclude that the function 0 and its second derivative must be constant multiples of each other. The only functions whose derivatives are constant multiples of the functions themselves are the exponential functions (or a linear combination of exponential functions such as sine and cosine hyperbolic functions). Therefore, the solution functions of the differential equation above are the exponential functions e or or constant multiples of them. This can be verified by direct substitution. For example, the second derivative of e is and its substitution into Eq. 3-56... 

The other hyperbolic trigonometric functions are the hyperbolic tangent, denoted by tanh(x) the hyperbolic cotangent, denoted by coth(x) the hyperbolic secant, denoted by sech(x) and the hyperbolic cosecant, denoted by csch(x). These functions are given by the equations... 

V. Numerical Values of hyperbolic functions.—Table IY. (pages 616, 617, and 618) contains numerical values of the hyperbolic sines and cosines for values of from 0 to 5, at intervals of 0 01. They have been checked by comparison with Des Ingenieurs Taschenbuch, edited by the Hiitte Academy, Berlin, 1877. The tables are used exactly like ordinary logarithm tables. Numerical values of the other functions can be easily deduced from those of sinh and cosh by the aid of equations (4). 

When an isothermal index value is required at a temperature other than the temperature it was measured at it can be obtained by interpolation using several empirical relationships like the hyperbolic function... 

When [SJo the enzyme concentration, Vq is usually directly proportional to the enzyme concentration in the reaction mixture, and for most enzymes Vq is a rectangular hyperbolic function of [S]q (see Fig. 5-15). If there are other (co-) substrates, then these are usually held constant during the series of experiments in which [SJo is varied. 

In the ladder model treatment of Blizard, an alternative termination of the line of springs (Fig. 10-3) was considered in which each end, rather than being fixed, is attached to three other such lines, each of these to three more, and so on indefinitely, thus reproducing the connectivity of a tetrafunctional network. This.proyision increases the equilibrium compliance by a factor of 2 (corresponding-fo the factor of (/ — 2)//mentioned in Section I above), and it modifies the frequency dependence, which is now expressed by a rather complicated combination of hyperbolic functions. This frequency dependence of J" is also shown in Fig. 10-7 the maximum is slightly broader than for fixed cross-links (i.e., cross-links with affine deformation). 

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

The most commonly used semiempirical for describing PES s is the diatomics-in-molecules (DIM) method. This method uses a Hamiltonian with parameters for describing atomic and diatomic fragments within a molecule. The functional form, which is covered in detail by Tully, allows it to be parameterized from either ah initio calculations or spectroscopic results. The parameters must be fitted carefully in order for the method to give a reasonable description of the entire PES. Most cases where DIM yielded completely unreasonable results can be attributed to a poor fitting of parameters. Other semiempirical methods for describing the PES, which are discussed in the reviews below, are LEPS, hyperbolic map functions, the method of Agmon and Levine, and the mole-cules-in-molecules (MIM) method. 

Inverse Laplace transforms have been tabulated for most analytical functions, including power, exponential, trigonometric, hyperbolic and other functions. In this context we require only the inverse Laplace transform which yields a simple exponential ... 

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