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Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. It also occurs in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

The ISO 80000-2 standard abbreviations consist of ar- followed by the abbreviation of the corresponding hyperbolic function (e.g., arsinh, arcosh).The prefix arc- followed by the corresponding hyperbolic function (e.g., arcsinh, arccosh) is also commonly seen, by analogy with the nomenclature for inverse trigonometric functions. These are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area; the hyperbolic functions are not directly related to arcs.[9][10][11]

Since the hyperbolic functions are rational functions of ex whose numerator and denominator are of degree at most two, these functions may be solved in terms of ex, by using the quadratic formula; then, taking the natural logarithm gives the following expressions for the inverse hyperbolic functions.

For complex arguments, the inverse hyperbolic functions, the square root and the logarithm are multi-valued functions, and the equalities of the next subsections may be viewed as equalities of multi-valued functions.

For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function. However, in some cases, the formulas of § Definitions in terms of logarithms do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected.

In the following graphical representation of the principal values of the inverse hyperbolic functions, the branch cuts appear as discontinuities of the color. The fact that the whole branch cuts appear as discontinuities, shows that these principal values may not be extended into analytic functions defined over larger domains. In other words, the above defined branch cuts are minimal.

The area functions are the inverse functions of the hyperbolic functions, i.e., the inverse hyperbolic functions. The functions sinh x, tanh x, and coth x are strictly monotone, so they have unique inverses without any restriction; the function cosh x has two monotonic intervals so we can consider two inverse functions. The name area refers to the fact that the geometric definition of the functions is the area of certain hyperbolic sectors ...

In math, there exist certain even and odd combinations of the natural exponential functions and which appear so frequently that they earned themselves their own special names. They are, in many ways, analogous to the trigonometric functions. For instance, they share the same relationship to the hyperbola that the trigonometric functions have to the circle. Because of this, these special functions are called hyperbolic functions.

The hyperbolic functions are essentially the trigonometric functions of the hyperbola. They extend the notion of the parametric equations for the unit circle, where , to the parametric equations for the unit hyperbola, and are defined in terms of the natural exponential function (where is Euler's number), giving us the following two fundamental hyperbolic formulas:

Beware! While the values of the derivatives are the same as with the trigonometric functions, the signs for the derivatives of hyperbolic cosine and hyperbolic secant are opposite to their trigonometric counterparts.

Notice that the inverse hyperbolic functions all involve logarithmic functions. This is because the hyperbolic functions involve exponential functions, and exponential and logarithmic functions are inverses of each other!

Hyperbolic functions are written similarly to the trigonometric functions for a circle. The six main hyperbolic functions are written as:sinh = (ex - e-x)/2cosh = (ex + e-x)/2tanh = (sinh x)/(cosh x)csch = 1/(sinh x)sech = 1/(cosh x)coth = (cosh x)/(sinh x)

respectively are characterized for various $t\,>\,0$ in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Lévy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for $t\,=\,1\,\,\text{or}\,2$ in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of ${{C}_{1}}\,\text{and}\,{{S}_{2}}$ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet $L$-function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from ${{S}_{t}}\,\text{and}\,{{C}_{t}}$ by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

Super-resolution microscopy, or nanoscopy, has become a critical tool for cell biology1, neuroscience2, pharmaceutical industry3, and nanophotonic applications4. The widely used super-resolution nanoscopy techniques are mostly based on fluorescence5,6,7, which have the capability to selectively label the targets on biological specimens. This fluorescent probe-based method provides exquisite image contrast resulting from high emission intensity contrast between the bright versus dark states of fluorophores; however, the restrictions in labeling protocols and alteration in cellular functions limit the range of imaging specimens8. Therefore, there is an urgent need to develop label-free super-resolution microscopy technologies.

Nevertheless, in contrast to fluorescence super-resolution nanoscopies, the resolution of the current super-resolution scattering imaging techniques is limited by the substrate materials that support a high effective refractive index at the desired wavelength. On one hand, dielectric materials have a limited refractive index. For example, super-resolution scattering image with a ~3-fold resolution improvement has been obtained by using GaP17 and it is hard to find any other materials which are better than GaP. On the other hand, plasmonic materials have shown much higher effective refractive indices at visible frequencies; therefore, super-resolution fluorescence microscopies based on plasmonic materials ranging from surface plasmon polariton structures18,19,20 to hyperbolic metamaterials have recently been theoretically proposed and experimentally demonstrated21,22,23. However, these plasmonic materials have only been applied to fluorescence imaging based on our knowledge. As a result, current label-free scattering imaging techniques have only demonstrated a resolution improvement of ~3 fold24,25,26,27.

A hyperbolic function is similar to a function but might differ to it in certain terms. The basic hyperbolic functions are hyperbola sin and hyperbola cosine from which the other functions are derived.

Hyperbolic functions are a special class of transcendental functions, similar to trigonometric functions or the natural exponential function, ex. Although not as common as their trig counterparts, the hyperbolics are useful for some applications, like modeling the shape of a power line hanging between two poles.

Unlike their trigonometric counterparts, hyperbolic functions are defined in terms of the exponential function ex. For example, f(x) = cosh(x) is defined by:And sinh(x) is defined as:All of the remaining hyperbolic functions (see list below) can be defined in terms of these two definitions.

The hyperbolic functions have similar names to the trigonometric functions, but they are defined in terms of the exponential function. The general trigonometric equations are defined using a circle. Hyperbolic functions are a set of trigonometric equations that are defined using a hyperbola rather than a circle.

Hyperbolic functions may be used to define a measure of distance in certain kinds of non-Euclidean geometry. For example, the hyperbolic cosine function may be used to describe the shape of the curve formed by a high-voltage line suspended between two towers. If you take a rope, fix the two ends, and let it hang under the force of gravity, it will naturally form a hyperbolic cosine curve. Because of its wide presence in the natural world, hyperbolic functions are an important part of physics.

The inverse hyperbolic functions, sometimes also called the area hyperbolic functions are the multivalued function that are the inverse functions of the hyperbolic functions. They are denoted \(cosh^(-1)x, coth^(-1)x, cosech^(-1)x, sech^(-1)x, sinh^(-1)x,\) and \(tanh^(-1)x\). 2b1af7f3a8