Hyperbolic functions, table

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). 

Hyperbolic functions are combinations of exponentials. They are given in Table A1.4, and these functions are plotted in Fig. A1.4. Since they are continuous functions, with continuous derivatives obtained in the same way as normal trigonometric functions, that is... 

MATHEMATICAL TABLES AND FORMULAS, Robert D. Camrichael and Edwin R. Smith. Logarithms, sines, tangents, trig functions, powers, roots, reciprocals, exponential and hyperbolic functions, formulas and theorems. 269pp. 55x854. 60111-0 Pa. 5.95... 

Figure 8.2 The relationship between total DON release, which is defined as the sum of DON released as a result of hoth NH4" and NO3 uptake, and NH4" regeneration in coastal and ocean systems. Model 11 regression parameters are reported in Table 8.7. The line for a hyperbolic function yields the following parameters y = 9.429Ln(x), = 0.53, n — 56. The graph includes data from...

Equations (1.80) and (1.81) provide relationships between complex variables and trigonometric functions. These can be manipulated to find relationships with hyperbolic function. Some important definitions and identities are presented in Table 1.6. ... 

TABLE 2-156 Heat Capacity at Constant Pressure of Inorganic and Organic Compounds in the Ideal Gas State Fit to Hyperbolic Functions Cp [J/(kmol K)]... 

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 —... 

Another substantial body of work dealing with cracking reactions has been reported over the past several years by Wojciechow-ski and coworkers. Results with cumene cracking over La-Y zeolite catalyst are summarized in entry 9 of Table 3. Catalyst deactivation in these studies is also treated as separable, but an hyperbolic function of time on stream is used for correlation of activity. The use of this type of correlation for a number of applications was summarized in a 1974 review (70). Since that time, in addition to recent work on cumene cracking (69) the model has been applied to an extensive series of studies on the cracking of gas oil distillates (71,72,73,74), as well as being employed in the correlation of Jacobs, et al. (77). 

Table 8.4 contains the deactivation parameters estimated from the long-term catalyst stability test and the feedstock evaluation with HCO. d and d are the parameters of the hyperbolic function that describes the initial activity decay caused by coke formation, whereas 3 is the exponent of the power-type function that represents the slow deactivation process by metal deposition (see Equation 8.21). Each set of... 

Table 2.1 Some Useful Relationships Involving the Hyperbolic Sine and Inverse Hyperbolic Sine Function...

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. 

Before training the net, the transfer functions of the neurons must be established. Here, different assays can be made (as detailed in the previous sections), but most often the hyperbolic tangent function tansig function in Table 5.1) is selected for the hidden layer. We set the linear transfer function purelin in Table 5.1) for the output layer. In all cases the output function was the identity function i.e. no further operations were made on the net signal given by the transfer function). 

In Figure 4 the overall polymerization time is plotted as a function of one or the other active species concentration. A hyperbolic type dependence of tp on [active species] is evident, with a very sharp decrease of tp in the concentration range between 0.3 and 0.7 mole %, and a much slower decrease at higher concentrations. At the highest levels of active species concentrations,tp is very low (ca. 3 min) and this value compares rather well with the usual reaction times for the RIM technology. Non-equimolar concentration conditions roughly follow the same pattern, as evidenced from the data quoted in Table V, and allow to underline the prominent role of [I] on tp, whereas [A] has a much lower relevance on it. 

The fundamental postulate of the time on stream theory is that the activity of the catalyst in a given reaction is purely a function of time. There are different types of catalyst decay according to this theory. The decay may be linear, exponential, hyperbolic and power function as given in Table 7.2. 

Previous stopped-flow fluorescence assays investigating matched dNTP incorporation showed that both the fast and the slow fluorescence transitions demonstrated a hyperbolic dependence on dNTP concentra-tion. " " " Similarly, the dNTP dependence of both the fast and the slow fluorescence phases during mismatched dNTP incorporation in stopped-flow has been examined. The observed rate constants for the fast and the slow phases, individually plotted as a function of dNTP concentration, reveal that both phases demonstrate a hyperbolic dependence on dNTP concentration (parameters obtained for k2, K, k o, and d,app as described in Section 8.10.4.2.3 and reported in Table 1). The observed hyperbolic dependence of the fast phase on mismatched dNTP largely indicates that this phase originates from a conformational change induced by mismatched dNTP binding. 

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