Sine hyperbolic

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. 

SBS styrene-butadiene-styrene SINEH sine hyperbolic... 

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

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 ... 

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)... 

It is readily seen that when p is large enough and the hyperbolic sines in (4.18) can be replaced by exponents, the effect of the prefactor B, is to replace the potential V s) by the vibrationally adiabatic... 

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/. 

Hyperbolic Functions sinh z = (e2 — e )/2 cosh z = e + 2)/2 tanh z = sinh z/cosh z coth z = cosh z/sinh z csch z = 1/sinh z sech z = 1/coshz. Identities are cosh2z - sinh2z = 1 sinh (z, + z2) = sinh z, cosh z2 + cosh Zi sinh z2 cosh (z, + z2) = cosh Zi cosh z2 + sinh Zi sinh z2 cosh z + sinh z =et cosh z - sinh z = e. The hyperbolic sine and hyperbolic cosine are periodic functions with the imaginary period 27ti. That is, sinh (z + 2iti) = sinh z. 

The hyperbolic sine function also fits many experimental simple tension data, and it has considerable theoretical foundation (77-88) ... 

These results have important consequences. By linearizing the hyperbolic sine function75... 

The hyperbolic sine and cosine functions sinh - and cosh. v are defined in terms of the sum and difference of the exponential functions e and e. respectively ... 

The quantity 2Na is the thickness, L, of the oxide layer. Also, the factor involving the difference in the exponentials can be expressed as twice the hyperbolic sine function. We therefore obtain... 

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... 

The coefficients in the above series in if/p(r) alternate between the hyperbolic sine and cosine value of the uniform contribution, i/r0(z). In contrast to a full linear treatment, which is the usual procedure followed, the 0(if/p) term here does not vanish. As if/0(z) is large we must regard it as satisfying the nonlinear, ordinary differential form of the PB equation,... 

---