Inverse hyperbolic functions

Inverse Hyperbolic Functions If x = sinh y, then y is the inverse hyperbolic sine of x written y = sinh" x or arcsinh x. sinh" x = log x + + 1)... 

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... 

We could get the same result by treating sinh u exactly as we treated sin on page 48, using the reference formulae of page 611. For the inverse hyperbolic functions, let... 

The inverse hyperbolic functions (on suitably chosen domains) are the area-functions arcosh, arsinh, and artanh. They satisfy... 

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

Introduction of the inverse hyperbolic sine function encourages us to take Eq. (2.24) a bit further and derive an expression for 17 itself. Before continuing, let us remember the following ... 

Hyperbolic functions frequently appear in electrochemical problems, for instance in the inversion of Laplace transforms. An important example of the use of the cosh function is in the expression for the differential capacity of the electrolyte double layer following the Gouy-Chapman model (Chapter 3) which has a minimum value and is symmetric around this minimum—compare Fig. 3.6 with the cosh function in Fig. A1.4. 

Figure A2.1. Hyperbolic functions, (a) even [b) uneven functions. Limits eire indicated, c. Inverse functions...

A more general theoretical model [29], which results in Equation 11.38, provides a better description of an elastomeric stress-strain curve all the way up to fracture, by including the finite extensibilities of the chains. In this equation, 1 is the inverse Langevin function. " is a transcendental function which is defined by Equation 11.39. Coth is the hyperbolic cotangent function. The superscript of -1 represents the functional inversion (and not merely simple reciprocal) of the function in square brackets. 

With limited amount of Ab, the unlabeled antigen (analyte) competes with the labeled antigen Ag for limited binding sites. Bound fraction (AgAb) is separated from free (Ab), and the signal [Ag Ab] complex (the Ab fraction not occupied by the analyte) is measured. The amount of analyte is inversely proportional to the bound [Ag Ab] complex in a hyperbolic function as in Fig. 1. Methods for transforming or linearizing these functions are presented in the section on data reduction (Sec. V). 

In the following formulas u, v, w represent functions of x, while a, c, n represent fixed real numbers. All arguments in the trigonometric functions are measured in radians, and all inverse trigonometric and hyperbolic functions represent principal values Let y = f(x) and = f (x) define, respectively, a function and its derivative for any value x in their common domain. 

All angles are measured in radians, and inverse trigonometric and hyperbolic functions represent principal values, unless otherwise indicated. 

The Shift and Hyp button make the inverse circular functions (Fig. 10.17) and the hyperbolic functions (Fig. 10.18), respectively, available. 

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

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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