Hyperbolic Function

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					                       Hyperbolic Function
Hyperbolic Function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or
circular, functions.

The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also called
"asinh" or sometimes "arcsinh") band so on.

Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t,
sinh t) form the right half of the equilateral hyperbola.

Hyperbolic functions occur in the solutions of some important linear differential
equations, for example the equation defining a catenary, of some cubic equations,
and of Laplace's equation in Cartesian coordinates.

The latter is important in many areas of physics, including electromagnetic theory,
heat transfer, fluid dynamics, and special relativity.

The hyperbolic functions take real values for a real argument called a hyperbolic
angle.
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In complex analysis, they are simply rational functions of exponentials, and so are
meromorphic.

Hyperbolic functions were introduced in the 1760s independently by Vincenzo
Riccati and Johann Heinrich Lambert.

Riccati used Sc. and Cc. to refer to circular functions and Sh. and Ch. to refer to
hyperbolic functions.

Lambert adopted the names but altered the abbreviations to what they are today.
The abbreviations sh and ch are still used in some other

In mathematics, hyperbolic Functions are similar to the trigonometric Functions,
and they are defined in terms of the Exponential Function. Here we are going to
define three main hyperbolic functions.

We also talk about some hyperbolic functions identities involving these functions,
with their inverse functions and reciprocal functions.

The functions f (s) = cosh s and f (s) = sinh s in terms of the exponential function.

The function f (s) = tanh s in terms of cosh s and sinh s.

The Inverse Function means sinh−1 s, cosh−1 s and tanh−1 s and specifies their
domains.

The hyperbolic Trigonometry includes different hyperbolic functions. The
Trigonometric Functions expressed in the form of ex are hyperbolic trigonometric
functions.


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Let’s talk about the different types of hyperbolic functions:

Hyperbolic   of   sine of‘s’ = sin hs =
Hyperbolic   of   cosine of ‘s’ = cos h s =
Hyperbolic   of   Tangent of ‘s’ = tan h s =
Hyperbolic   of   cotangent of ‘s’ = cot h s =
Hyperbolic   of   tangent of ‘s’ = tan h s =

Hyperbolic of secant of ‘s’ = sec h s =

Hyperbolic of cosecant of ‘s’ = csc h s=

Now we will talk about the negative hyperbolic function which is given as:

Sin h (-s) = -sin h s,
Cos h (-s) = cos h s,
Tan h (-s) = -tan h s,
Cosec h (-s) = -cosec h s,
Sec h (-s) = sec h s,
Cot h (-s) = -cot h s,




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