Day 31

 "Hyperbolic curve" redirects here. For the geometric curve, see Hyperbola.

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. 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.

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur 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 basic hyperbolic functions are:^[1]

• hyperbolic sine "sinh" (/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/),^[2]
• hyperbolic cosine "cosh" (/ˈkɒʃ, ˈkoʊʃ/),^[3]

from which are derived:^[4]

• hyperbolic tangent "tanh" (/ˈtæŋ, ˈtæntʃ, ˈθæn/),^[5]
• hyperbolic cotangent "coth" (/ˈkɒθ, ˈkoʊθ/),^[6][7]
• hyperbolic secant "sech" (/ˈsɛtʃ, ˈʃɛk/),^[8]
• hyperbolic cosecant "csch" or "cosech" (/ˈkoʊsɛtʃ, ˈkoʊʃɛk/^[3])

corresponding to the derived trigonometric functions.

The inverse hyperbolic functions are:

• area hyperbolic sine "arsinh" (also denoted "sinh⁻¹", "asinh" or sometimes "arcsinh")^[9][10][11]
• area hyperbolic cosine "arcosh" (also denoted "cosh⁻¹", "acosh" or sometimes "arccosh")
• area hyperbolic tangent "artanh" (also denoted "tanh⁻¹", "atanh" or sometimes "arctanh")
• area hyperbolic cotangent "arcoth" (also denoted "coth⁻¹", "acoth" or sometimes "arccoth")
• area hyperbolic secant "arsech" (also denoted "sech⁻¹", "asech" or sometimes "arcsech")
• area hyperbolic cosecant "arcsch" (also denoted "arcosech", "csch⁻¹", "cosech⁻¹","acsch", "acosech", or sometimes "arccsch" or "arccosech")

A ray through the unit hyperbola x² − y² = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).

The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. 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. The hyperbolic sine and the hyperbolic cosine are entire functions. 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.^[13] Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. 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.

Notation

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Main article: Trigonometric functions § Notation

Definitions

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sinh, cosh and tanh

csch, sech and coth

There are various equivalent ways to define the hyperbolic functions.

Exponential definitions

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sinh x is half the difference of e^x and e^−x

cosh x is the average of e^x and e^−x

In terms of the exponential function:^[1][4]

• Hyperbolic sine: the odd part of the exponential function, that is,
  $${\displaystyle \sinh x=\frac{e^x-e^{-x}}{2}=\frac{e^{2x}-1}{2e^x}=\frac{1-e^{-2x}}{2e^{-x}}.}$$
• Hyperbolic cosine: the even part of the exponential function, that is,
  $${\displaystyle \cosh x=\frac{e^x+e^{-x}}{2}=\frac{e^{2x}+1}{2e^x}=\frac{1+e^{-2x}}{2e^{-x}}.}$$
• Hyperbolic tangent:
  $${\displaystyle \tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}=\frac{e^{2x}-1}{e^{2x}+1}.}$$
• Hyperbolic cotangent: for x ≠ 0,
  $${\displaystyle \coth x=\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\frac{e^{2x}+1}{e^{2x}-1}.}$$
• Hyperbolic secant:
  $${\displaystyle \mathrm{sech}x=\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}=\frac{2e^x}{e^{2x}+1}.}$$
• Hyperbolic cosecant: for x ≠ 0,
  $${\displaystyle \mathrm{csch}x=\frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}=\frac{2e^x}{e^{2x}-1}.}$$

Differential equation definitions

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The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the solution (s, c) of the system

with the initial conditions  The initial conditions make the solution unique; without them any pair of functions  would be a solution.

sinh(x) and cosh(x) are also the unique solution of the equation f ″(x) = f (x), such that f (0) = 1, f ′(0) = 0 for the hyperbolic cosine, and f (0) = 0, f ′(0) = 1 for the hyperbolic sine.

Complex trigonometric definitions

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Hyperbolic functions may also be deduced from trigonometric functions with complex arguments:

• Hyperbolic sine:^[1]
  $${\displaystyle \sinh x=-i\sin (ix).}$$
• Hyperbolic cosine:^[1]
  $${\displaystyle \cosh x=\cos (ix).}$$
• Hyperbolic tangent:
  $${\displaystyle \tanh x=-i\tan (ix).}$$
• Hyperbolic cotangent:
  $${\displaystyle \coth x=i\cot (ix).}$$
• Hyperbolic secant:
  $${\displaystyle \mathrm{sech}x=\sec (ix).}$$
• Hyperbolic cosecant:
  $${\displaystyle \mathrm{csch}x=i\csc (ix).}$$

where i is the imaginary unit with i² = −1.

The above definitions are related to the exponential definitions via Euler's formula (See § Hyperbolic functions for complex numbers below).

Characterizing properties

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Hyperbolic cosine

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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]

$${\displaystyle \text{area}={\int }_a^b\cosh xdx={\int }_a^b\sqrt{1+{\left(\frac{d}{dx}\cosh x\right)}^2}dx=\text{arc length.}}$$

Hyperbolic tangent

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The hyperbolic tangent is the (unique) solution to the differential equation f ′ = 1 − f ², with f (0) = 0.^[16][17]

Useful relations

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The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. 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 ${\displaystyle \theta }$, ${\displaystyle 2\theta }$, ${\displaystyle 3\theta }$ or ${\displaystyle \theta }$ and ${\displaystyle \phi }$ 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.

Odd and even functions:

Hence:

Thus, cosh x and sech x are even functions; the others are odd functions.

Hyperbolic sine and cosine satisfy:

the last of which is similar to the Pythagorean trigonometric identity.

One also has

for the other functions.

Sums of arguments

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particularly

Also:

Subtraction formulas

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Also:^[19]

Half argument formulas

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where sgn is the sign function.

If x ≠ 0, then^[20]

$${\displaystyle \tanh \left(\frac{x}{2}\right)=\frac{\cosh x-1}{\sinh x}=\coth x-\mathrm{csch}x}$$

Square formulas

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Inequalities

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The following inequality is useful in statistics:^[21]

$${\displaystyle \cosh (t)\le e^{t^2/2}.}$$

It can be proved by comparing the Taylor series of the two functions term by term.

Inverse functions as logarithms

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Main article: Inverse hyperbolic function

Derivatives

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Second derivatives

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Each of the functions sinh and cosh is equal to its second derivative, that is:

$${\displaystyle \frac{d^2}{d{x}^2}\sinh x=\sinh x}$$$${\displaystyle \frac{d^2}{d{x}^2}\cosh x=\cosh x.}$$

All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions ${\displaystyle e^x}$ and ${\displaystyle e^{-x}}$.^[22]

Standard integrals

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For a full list, see list of integrals of hyperbolic functions.

The following integrals can be proved using hyperbolic substitution:

where C is the constant of integration.

Taylor series expressions

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It is possible to express explicitly the Taylor series at zero (or the Laurent series, if the function is not defined at zero) of the above functions.

$${\displaystyle \sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{x^{2n+1}}{(2n+1)!}}$$

This series is convergent for every complex value of x. Since the function sinh x is odd, only odd exponents for x occur in its Taylor series.

$${\displaystyle \cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\cdots =\sum _{n=0}^{\mathrm{\infty }}\frac{x^{2n}}{(2n)!}}$$

This series is convergent for every complex value of x. Since the function cosh x is even, only even exponents for x occur in its Taylor series.

The sum of the sinh and cosh series is the infinite series expression of the exponential function.

The following series are followed by a description of a subset of their domain of convergence, where the series is convergent and its sum equals the function.

where:

• ${\displaystyle B_n}$ is the nth Bernoulli number
• ${\displaystyle E_n}$ is the nth Euler number

Infinite products and continued fractions

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The following expansions are valid in the whole complex plane:

${\displaystyle \sinh x=x\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{x^2}{n^2{\pi }^2}\right)=\frac{x}{1-\frac{x^2}{2\cdot 3+x^2-\frac{2\cdot 3x^2}{4\cdot 5+x^2-\frac{4\cdot 5x^2}{6\cdot 7+x^2-\ddots }}}}}$

${\displaystyle \cosh x=\prod _{n=1}^{\mathrm{\infty }}\left(1+\frac{x^2}{(n-1/2{)}^2{\pi }^2}\right)=\frac{1}{1-\frac{x^2}{1\cdot 2+x^2-\frac{1\cdot 2x^2}{3\cdot 4+x^2-\frac{3\cdot 4x^2}{5\cdot 6+x^2-\ddots }}}}}$

${\displaystyle \tanh x=\frac{1}{\frac{1}{x}+\frac{1}{\frac{3}{x}+\frac{1}{\frac{5}{x}+\frac{1}{\frac{7}{x}+\ddots }}}}}$

Comparison with circular functions

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Circle and hyperbola tangent at (1,1) display geometry of circular functions in terms of circular sector area u and hyperbolic functions depending on hyperbolic sector area u.

The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle.

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. Similarly, the yellow and red regions together depict a hyperbolic sector with area corresponding to hyperbolic 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.^[23]

The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.

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.

Relationship to the exponential function

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The decomposition of the exponential function in its even and odd parts gives the identities

$${\displaystyle e^x=\cosh x+\sinh x,}$$

and

$${\displaystyle e^{-x}=\cosh x-\sinh x.}$$

Combined with Euler's formula

$${\displaystyle e^{ix}=\cos x+i\sin x,}$$

this gives

$${\displaystyle e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)}$$

for the general complex exponential function.

Additionally,

$${\displaystyle e^x=\sqrt{\frac{1+\tanh x}{1-\tanh x}}=\frac{1+\tanh \frac{x}{2}}{1-\tanh \frac{x}{2}}}$$