Hyperbolic functions, relations

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

The inverse hyperbolic functions, sinh" x, etc., are related to the logarithmic functions and are particularly useful in integral calculus. These relationships may be defined for real numbers x and y as... 

Because of this duality, every relation involving circular functions btf formal counterpart in foe corresponding hyperbolic functions, and vice ... 

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

The three experimental parameters A, B and C are related to column parameters and also to experimental conditions. If H is expressed in cm, A will be expressed in cm, B in cm2/s and C in s (where velocity is measured in cm/s). The function is a hyperbolic function that goes through a minimum (//mjn) when ... 

The relation between the volumetric weight and the mean cell diameter is described by a hyperbolic function. This function has been analytically derived by Romanenkov et al. from the following geometrical model. 

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. 

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

The first term In equation (14), representing the gross rate of biomass production, Is Identical with the function Monod (25) originally adopted "to express conveniently the relation between exponential growth rate and concentration of an essential nutrient." Such a rectangular hyperbolic function has been derived many times from various reaction mechanisms (26-30). but none has addressed the present case of continuous culture systems where y j and K have been observed to vary with temperature and dilution rate. 

The determinant of coefficients of these two equations yields, as before, a relation that can be used to calculate the time factor p. We have already assumed a/t 1 in writing Equation 5.110. If we further assume lar /t l 1, we can expand the hyperbolic functions and obtain a result first given by Vrij et al. (1970) ... 

In these expressions, the relations (5.215) have been used as the changing of variables, along the connection between the hyperbolic functions and the series of the modified Bessel functions while for the integrals calcula-... 

The difference in exponentials which occurs in Eq. (2.21) is directly related to the hyperbolic sine function... 

The trigonometric functions developed in the previous seefidtiare lef pd to as circidar functions, as they are related to the circle shown in Fig, 11, somewhat less familiar family of functions, the hyperbolic funefens, c also be derived from the exponential. They are analogous to the circular iom considered above and can be defined bv the relations... 

Vol. 1539 M. Coomaert, A. Papadopoulos, Symbolic Dynamics and Hyperbolic Groups. VIII, 138 pages. 1993. Vol. 1540 H. Komatsu (Ed.), Functional Analysis and Related Topics, 1991. Proceedings. XXI, 413 pages. 1993. 

---