Hyperbolic functions
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 unit hyperbola.
Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.
They are used to express Lorentz boosts as hyperbolic rotations in special relativity.
Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics.
The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.
In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle.
As a result, the other hyperbolic functions are meromorphic in the whole complex plane.
By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.
[12] Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.
Lambert adopted the names, but altered the abbreviations to those used today.
[14] The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.
There are various equivalent ways to define the hyperbolic functions.
In terms of the exponential function:[1][4] The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the solution (s, c) of the system
The initial conditions make the solution unique; without them any pair of functions
It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the arc length corresponding to that interval:[15]
The hyperbolic tangent is the (unique) solution to the differential equation f ′ = 1 − f 2, with f (0) = 0.
In fact, Osborn's rule[18] states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for
into a hyperbolic identity, by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term containing a product of two sinhs.
Each of the functions sinh and cosh is equal to its second derivative, that is:
Since the function cosh x is even, only even exponents for x occur in its Taylor series.
where: The following expansions are valid in the whole complex plane: The hyperbolic functions represent an expansion of trigonometry beyond the circular functions.
Since the area of a circular sector with radius r and angle u (in radians) is r2u/2, it will be equal to u when r = √2.
In the diagram, such a circle is tangent to the hyperbola xy = 1 at (1,1).
The yellow sector depicts an area and angle magnitude.
The legs of the two right triangles with hypotenuse on the ray defining the angles are of length √2 times the circular and hyperbolic functions.
The hyperbolic angle is an invariant measure with respect to the squeeze mapping, just as the circular angle is invariant under rotation.
The graph of the function a cosh(x/a) is the catenary, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
The decomposition of the exponential function in its even and odd parts gives the identities
Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments.
Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:
