This is done as relatively rapid changes in indifference points are observed at relatively shorter delay values, whereas changes become less pronounced out at relatively longer delay values, which is characteristic of

hyperbolic functions.

Hyperbolic models have also been used in geomechanics problems to model interfaces [17, 18], soil reinforcement pullout [19], and triaxial compression tests on soil [20], It should be noted that

hyperbolic functions have provided components of tensile load-strain models reported in the geosynthetics literature [1, 21-25].

Right and skew-right circulant matrices remains invariant on their type when evaluated on trigonometric and

hyperbolic functions that are used in this paper.

Later chapters cover continuity of functions, derivatives of functions, trigonometric limits, and exponential and logarithmic functions and their derivatives, in addition to implicit functions, parametric functions, Rolle's theorem, Taylor's formula, and

hyperbolic functions.

We applied deform

hyperbolic functions introduced for the first time by Arai in (ARAI, 1991)

There's also an advanced math library, including trigonometry and

hyperbolic functions.

Keywords: Pensions, Retirement,

Hyperbolic Functions, Consumption, Saving, Time Preference.

He included appendices on the D-operator methods and

hyperbolic functions and ODES.

The complement of processing commands for brightness and contrast includes stretching routines with linear, nonlinear, and

hyperbolic functions.

One disadvantage of the

hyperbolic function is that age-price functions estimated from a

hyperbolic function (or alternatively,

hyperbolic functions estimated from an age-price function) require an assumption to be made about a real discount rate.

To derive |Beta~l, A = cosh(|Gamma~l) in the ABCD matrix can be expanded using an identity for

hyperbolic functions as

Similarly, the development of the

hyperbolic functions follows from the unit equilateral hyperbola, [x.