Hyperbolic function

\scriptstyle x^{2}\ -\ y^{2}\ =\ 1

in the point

\scriptstyle (\cosh \,a,\,\sinh \,a)

, where

\scriptstyle a

is twice the area between the ray, the hyperbola, and the

\scriptstyle x

-axis. For points on the hyperbola below the

\scriptstyle x

-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).

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

The basic hyperbolic functions are the hyperbolic sine "sinh" (/ˈsɪntʃ/ or /ˈʃaɪn/),^[1] and the hyperbolic cosine "cosh" (/ˈkɒʃ/),^[2] from which are derived the hyperbolic tangent "tanh" (/ˈtæntʃ/ or /ˈθæn/),^[3] hyperbolic cosecant "csch" or "cosech" (/ˈkoʊʃɛk/^[2] or /ˈkoʊsɛtʃ/), hyperbolic secant "sech" (/ˈʃɛk/ or /ˈsɛtʃ/),^[4] and hyperbolic cotangent "coth" (/ˈkoʊθ/ or /ˈkɒθ/),^[5]^[6] corresponding to the derived trigonometric functions.

The inverse hyperbolic functions are the area hyperbolic sine "arsinh" (also denoted "sinh⁻¹", "asinh" or sometimes "arcsinh")^[7]^[8]^[9] and 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. 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.

Hyperbolic functions occur in the solutions of many linear differential equations, for example the equation defining a catenary, of some cubic equations, in calculations of angles and distances in hyperbolic geometry and of 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.

In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. When considered defined by a complex variable, the hyperbolic functions are rational functions of exponentials, and are hence holomorphic.

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.^[10] Riccati used Sc. and Cc. ([co]sinus circulare) to refer to circular functions and Sh. and Ch. ([co]sinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today.^[11] The abbreviations sh and ch are still used in some other languages, like French and Russian.

Definitions

Defined on a unit hyperbola

Hyperbolic functions are defined on a hyperbola similar to the way circular trigonometric functions are defined on a circle. To see the similarities, first consider a circle of radius one $(r=1)$ centered on the origin of a Cartesian coordinate system. The equation of this unit circle is given by

Figure 1 One way of defining the circular trigonometric functions is on a circle with a radius of 1.

x^{2}+y^{2}=r^{2}=1.

The argument of the circular trigonometric functions is usually expressed as an angle like theta in Figure 1. An angle (in radians) is defined on a circle and is equal to the subtended arc length $s$ divided by the radius of the circle. Therefore, on the unit circle, an angle and its subtended arc length are numerically equal. But notice that the circumference of the unit circle is $2\pi$ and the area is $\pi$ . Therefore, an angle on a unit circle is numerically equal to twice the area of the sector subtended by that angle (the red area in Figure 1). So, for a point P we could define the two fundamental trig functions using an area on the unit circle instead of an angle. For example

\sin \theta =\sin(2A)={\frac {\textrm {ordinate}}{\textrm {radius}}}={\frac {y}{r}}=y

\cos \theta =\cos(2A)={\frac {\textrm {abcissa}}{\textrm {radius}}}={\frac {x}{r}}=x.

Note that a circle is an ellipse, but since its semi major axis and semi minor axis are equal, they are called the radius. Euler showed that

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

(Euler’s formula)

and therefore

e^{-ix}=\cos x-i\sin x.

Adding and then subtracting the two equations yields

$\cos x={\frac {e^{ix}+e^{-ix}}{2}}$

(1)

$\sin x={\frac {e^{ix}-e^{-ix}}{2i}}.$

(2)

Since they are equivalent, we could define the sine and cosine using exponentials as in equations (1) and (2) or equivalently we could define them geometrically on a circle. Hyperbolic functions can also be expressed as exponentials or geometrically on a hyperbola. Just as the area of a sector on the unit circle can be used to define the circular functions, an area is used as the argument of the hyperbolic functions. One way of defining hyperbolic functions is on the positive $x$ branch of a unit hyperbola given by the equation

Figure 2 One way of defining the hyperbolic functions is on the positive branch of a unit hyperbola.

x=+{\sqrt {y^{2}+1}}.

A ray drawn from the origin $O$ to a point P on the hyperbola defines an area (called a hyperbolic sector) bounded by the ray, the $x$ -axis, and the hyperbola (the red area in Figure 2). Following the procedure used for the unit circle, the hyperbolic sine and cosine are defined as

\sinh(2A)=\sinh u={\frac {\mbox{ordinate}}{\mbox{semi major axis}}}={\frac {y}{a}}=y

(sinh is pronounced “cinch”)

\cosh(2A)=\cosh u={\frac {\mbox{abcissa}}{\mbox{semi major axis}}}={\frac {x}{a}}=x.

(cosh rhymes with “gosh”)

where $u$ is twice the area of the hyperbolic sector shown in Figure 2. Note that $u$ is not related to any angle in the figure in a simple way as the area of the sector in the circle was. The $\sinh u$ and $\cosh u$ are the coordinates of point P on the unit hyperbola (Figure 2), just as $\sin \theta$ and $\cos \theta$ are the coordinates of point P on the unit circle (Figure 1). Areas below the $x$ -axis are considered negative, so from our definitions we can see that

\sinh u=-\sinh {(-u)}

\cosh u=\cosh {(-u)}.

The area of the sector $A$ can be obtained by integration. $A$ is the area of the right triangle with legs $\cosh u=x$ and $\sinh u=y$ minus the area under the positive $y$ part of the hyperbola between 1 and $\cosh u=x$ . The equation for the positive $y$ part of the hyperbola is

$y=+{\sqrt {x^{2}-1}}.$

(3)

Thus the area is

$A={\frac {1}{2}}xy-\int _{1}^{x}y\,dx={\frac {1}{2}}xy-\int _{1}^{x}{\sqrt {x^{2}-1}}\,dx={\frac {1}{2}}\ln(x+y)$

(4)

and

u=2A=\ln(x+y)=\ln(\cosh u+\sinh u).

Taking the antilog gives an equation similar to Euler’s formula, but for hyperbolic functions.

e^{u}=\cosh u+\sinh u

and replacing $u$ by $(-u)$ gives

e^{-u}=\cosh u-\sinh u.

By adding and then subtracting the last two equations we get

\cosh u={\frac {e^{u}+e^{-u}}{2}}

\sinh u={\frac {e^{u}-e^{-u}}{2}}

which are similar to equations (1) and (2) for circular functions. Therefore, the exponential representations of the two basic hyperbolic functions are equivalent to their geometrical definitions on the unit hyperbola. Theta, the argument of the circular trig functions is an angle, but $u$ , the argument of the hyperbolic functions is an area. The angle phi that defines the red area in Figure 2 is related to that area, but in a rather complicated way. Here is where the analogy to circular functions breaks down. From Figure 2, we see that

\tan \phi ={\frac {\sinh u}{\cosh u}}={\frac {y}{x}}.

Figure 3 Graphs of the hyperbolic sine, cosine, and tangent plotted as functions of the area

u

By using this tangent equation and equation (3) to eliminate $x$ and $y$ from equation (4) we obtain

A={\frac {1}{2}}\ln {\frac {1+\tan \phi }{\sqrt {1-\tan ^{2}\phi }}}.

Notice that as the angle phi approaches ${\frac {\pi }{4}},\tan \phi$ approaches one, and the area $A$ becomes infinite. Sometimes to confuse the issue, the argument of the hyperbolic functions is called the hyperbolic angle. Obviously a poor choice since $u$ is an area, not an angle, and furthermore it is not related to an angle in any simple way.

From the two fundamental hyperbolic functions, the other derived functions can be defined.

\tanh u={\frac {\sinh u}{\cosh u}}

(tanh rhymes with “ranch”)

etc.

Graphs of the hyperbolic sine, cosine, and tangent as a function of $u$ are shown in Figure 3. Of course, since $u$ is just a number, you can relabel the argument and call it $x$ if you like. This was done in the following sections.

Other definitions

sinh, cosh and tanh

csch, sech and coth

(a) cosh(x) is the average of e^x and e^−x

(b) sinh(x) is half the difference of e^x and e^−x

There are various equivalent ways for defining the hyperbolic functions. They may be defined in terms of the exponential function:

Hyperbolic sine:

\sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}.

Hyperbolic cosine:

\cosh x={\frac {e^{x}+e^{-x}}{2}}={\frac {e^{2x}+1}{2e^{x}}}={\frac {1+e^{-2x}}{2e^{-x}}}.

Hyperbolic tangent:

\tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\frac {e^{2x}-1}{e^{2x}+1}}={\frac {1-e^{-2x}}{1+e^{-2x}}}.

Hyperbolic cotangent:

\coth x={\frac {\cosh x}{\sinh x}}={\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\frac {e^{2x}+1}{e^{2x}-1}}={\frac {1+e^{-2x}}{1-e^{-2x}}},\qquad x\neq 0.

Hyperbolic secant:

\operatorname {sech} \,x={\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}={\frac {2e^{x}}{e^{2x}+1}}={\frac {2e^{-x}}{1+e^{-2x}}}.

Hyperbolic cosecant:

\operatorname {csch} \,x={\frac {1}{\sinh x}}={\frac {2}{e^{x}-e^{-x}}}={\frac {2e^{x}}{e^{2x}-1}}={\frac {2e^{-x}}{1-e^{-2x}}},\qquad x\neq 0.

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

{\begin{aligned}c'(x)&=s(x)\\s'(x)&=c(x)\end{aligned}}

such that $s (0) = 0$ and $c (0) = 1$ .

They 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.

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

Hyperbolic sine:

\sinh x=-i\sin(ix)

Hyperbolic cosine:

\cosh x=\cos(ix)

Hyperbolic tangent:

\tanh x=-i\tan(ix)

Hyperbolic cotangent:

\coth x=i\cot(ix)

Hyperbolic secant:

\operatorname {sech} x=\sec(ix)

Hyperbolic cosecant:

\operatorname {csch} x=i\csc(ix)

where i is the imaginary unit with the property that $i^{2}=-1.$

The complex forms in the definitions above derive from Euler's formula.

Characterizing properties

Hyperbolic cosine

It can be shown that the area under the curve of cosh (x) over a finite interval is always equal to the arc length corresponding to that interval:^[12]

{\text{area}}=\int _{a}^{b}{\cosh {(x)}}\ dx=\int _{a}^{b}{\sqrt {1+\left({\frac {d}{dx}}\cosh {(x)}\right)^{2}}}\ dx={\text{arc length}}

Hyperbolic tangent

The hyperbolic tangent is the solution to the differential equation $f'=1-f^{2}$ with f(0)=0 and the nonlinear boundary value problem:^[13] ^[14]

{\frac {1}{2}}f''=f^{3}-f;\quad f(0)=f'(\infty )=0

Useful relations

Odd and even functions:

{\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}

Hence:

{\begin{aligned}\tanh(-x)&=-\tanh x\\\coth(-x)&=-\coth x\\\operatorname {sech} (-x)&=\operatorname {sech} x\\\operatorname {csch} (-x)&=-\operatorname {csch} x\end{aligned}}

It can be seen that cosh x and sech x are even functions; the others are odd functions.

{\begin{aligned}\operatorname {arsech} x&=\operatorname {arcosh} {\frac {1}{x}}\\\operatorname {arcsch} x&=\operatorname {arsinh} {\frac {1}{x}}\\\operatorname {arcoth} x&=\operatorname {artanh} {\frac {1}{x}}\end{aligned}}

Hyperbolic sine and cosine satisfy:

{\begin{aligned}\cosh x+\sinh x&=e^{x}\\\cosh x-\sinh x&=e^{-x}\\\cosh ^{2}x-\sinh ^{2}x&=1\end{aligned}}

the last which is similar to the Pythagorean trigonometric identity.

One also has

{\begin{aligned}\operatorname {sech} ^{2}x&=1-\tanh ^{2}x\\\operatorname {csch} ^{2}x&=\coth ^{2}x-1\end{aligned}}

for the other functions.

Sums of arguments

{\begin{aligned}\sinh(x+y)&=\sinh(x)\cosh(y)+\cosh(x)\sinh(y)\\\cosh(x+y)&=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)\\\tanh(x+y)&={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\\\end{aligned}}

particularly

{\begin{aligned}\cosh(2x)&=\sinh ^{2}{x}+\cosh ^{2}{x}=2\sinh ^{2}x+1=2\cosh ^{2}x-1\\\sinh(2x)&=2\sinh x\cosh x\end{aligned}}

Also:

{\begin{aligned}\sinh x+\sinh y&=2\sinh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\\\cosh x+\cosh y&=2\cosh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\\\end{aligned}}

Subtraction formulas

{\begin{aligned}\sinh(x-y)&=\sinh(x)\cosh(y)-\cosh(x)\sinh(y)\\\cosh(x-y)&=\cosh(x)\cosh(y)-\sinh(x)\sinh(y)\\\end{aligned}}

Also:

{\begin{aligned}\sinh x-\sinh y&=2\cosh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\\\cosh x-\cosh y&=2\sinh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\\\end{aligned}}

Source.^[15]

Half argument formulas

\sinh \left({\frac {x}{2}}\right)={\frac {\sinh(x)}{\sqrt {2(\cosh(x)+1)}}}=\operatorname {sgn}(x)\,{\sqrt {\frac {\cosh(x)-1}{2}}}

where sgn is the sign function.

\cosh \left({\frac {x}{2}}\right)={\sqrt {\frac {\cosh(x)+1}{2}}}

\tanh \left({\frac {x}{2}}\right)={\frac {\sinh(x)}{\cosh(x)+1}}=\operatorname {sgn}(x)\,{\sqrt {\frac {\cosh(x)-1}{\cosh(x)+1}}}={\frac {e^{x}-1}{e^{x}+1}}

If x ≠ 0, then

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

^[16]

Inverse functions as logarithms

{\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geqslant 1\\\operatorname {artanh} (x)&={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)&&|x|<1\\\operatorname {arcoth} (x)&={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)&&|x|>1\\\operatorname {arsech} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}-1}}\right)=\ln \left({\frac {1+{\sqrt {1-x^{2}}}}{x}}\right)&&0<x\leqslant 1\\\operatorname {arcsch} (x)&=\ln \left({\frac {1}{x}}+{\sqrt {{\frac {1}{x^{2}}}+1}}\right)=\ln \left({\frac {1+{\sqrt {x^{2}+1}}}{x}}\right)&&x\neq 0\end{aligned}}

Derivatives

{\begin{aligned}{\frac {d}{dx}}\sinh x&=\cosh x\\{\frac {d}{dx}}\cosh x&=\sinh x\\{\frac {d}{dx}}\tanh x&=1-\tanh ^{2}x=\operatorname {sech} ^{2}x={\frac {1}{\cosh ^{2}x}}\\{\frac {d}{dx}}\coth x&=1-\coth ^{2}x=-\operatorname {csch} ^{2}x=-{\frac {1}{\sinh ^{2}x}}&&x\neq 0\\{\frac {d}{dx}}\operatorname {sech} x&=-\tanh x\operatorname {sech} x\\{\frac {d}{dx}}\ \operatorname {csch} \,x&=-\coth x\ \operatorname {csch} \,x&&x\neq 0\\{\frac {d}{dx}}\,\operatorname {arsinh} \,x&={\frac {1}{\sqrt {x^{2}+1}}}\\{\frac {d}{dx}}\,\operatorname {arcosh} \,x&={\frac {1}{\sqrt {x^{2}-1}}}&&1<x\\{\frac {d}{dx}}\,\operatorname {artanh} \,x&={\frac {1}{1-x^{2}}}&&|x|<1\\{\frac {d}{dx}}\,\operatorname {arcoth} \,x&={\frac {1}{1-x^{2}}}&&1<|x|\\{\frac {d}{dx}}\,\operatorname {arsech} \,x&=-{\frac {1}{x{\sqrt {1-x^{2}}}}}&&0<x<1\\{\frac {d}{dx}}\,\operatorname {arcsch} \,x&=-{\frac {1}{|x|{\sqrt {1+x^{2}}}}}&&x\neq 0\\\end{aligned}}

Second derivatives

Sinh and cosh are both equal to their second derivative, that is:

{\frac {d^{2}}{dx^{2}}}\sinh x=\sinh x\,

{\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.

All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions $e^{x}$ and $e^{-x}$ , and the zero function $f(x)=0$ .

Standard integrals

{\begin{aligned}\int \sinh(ax)\,dx&=a^{-1}\cosh(ax)+C\\\int \cosh(ax)\,dx&=a^{-1}\sinh(ax)+C\\\int \tanh(ax)\,dx&=a^{-1}\ln(\cosh(ax))+C\\\int \coth(ax)\,dx&=a^{-1}\ln(\sinh(ax))+C\\\int \operatorname {sech} (ax)\,dx&=a^{-1}\arctan(\sinh(ax))+C\\\int \operatorname {csch} (ax)\,dx&=a^{-1}\ln \left(\tanh \left({\frac {ax}{2}}\right)\right)+C=a^{-1}\ln \left|\operatorname {csch} (ax)-\coth(ax)\right|+C\end{aligned}}

The following integrals can be proved using hyperbolic substitution:

{\begin{aligned}\int {{\frac {1}{\sqrt {a^{2}+u^{2}}}}\,du}&=\operatorname {arsinh} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{\sqrt {u^{2}-a^{2}}}}\,du}&=\operatorname {arcosh} \left({\frac {u}{a}}\right)+C\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {artanh} \left({\frac {u}{a}}\right)+C&&u^{2}<a^{2}\\\int {\frac {1}{a^{2}-u^{2}}}\,du&=a^{-1}\operatorname {arcoth} \left({\frac {u}{a}}\right)+C&&u^{2}>a^{2}\\\int {{\frac {1}{u{\sqrt {a^{2}-u^{2}}}}}\,du}&=-a^{-1}\operatorname {arsech} \left({\frac {u}{a}}\right)+C\\\int {{\frac {1}{u{\sqrt {a^{2}+u^{2}}}}}\,du}&=-a^{-1}\operatorname {arcsch} \left|{\frac {u}{a}}\right|+C\end{aligned}}

where C is the constant of integration.

Taylor series expressions

It is possible to express the above functions as Taylor series:

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

The function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh(−x), and sinh 0 = 0.

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

The function cosh x has a Taylor series expression with only even exponents for x. Thus it is an even function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the infinite series expression of the exponential function.

{\begin{aligned}\tanh x&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}\\\coth x&=x^{-1}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots =x^{-1}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi \\\operatorname {sech} \,x&=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}\\\operatorname {csch} \,x&=x^{-1}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots =x^{-1}+\sum _{n=1}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi \end{aligned}}

where:

B_{n}\,

is the nth Bernoulli number

E_{n}\,

is the nth Euler number

Comparison with circular functions

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 is ${\frac {r^{2}u}{2}},$ it will be equal to u when r = square root of 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 sectors together depict an area and 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.^[17]

Identities

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 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 which contains a product of 2, 6, 10, 14, ... sinhs. This yields for example the addition theorems

{\begin{aligned}\sinh(x+y)&=\sinh(x)\cosh(y)+\cosh(x)\sinh(y)\\\cosh(x+y)&=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)\\\tanh(x+y)&={\frac {\tanh(x)+\tanh(y)}{1+\tanh(x)\tanh(y)}}\end{aligned}}

the "double argument formulas"

{\begin{aligned}\sinh(2x)&=2\sinh x\cosh x\\\cosh(2x)&=\cosh ^{2}x+\sinh ^{2}x=2\cosh ^{2}x-1=2\sinh ^{2}x+1\\\tanh(2x)&={\frac {2\tanh x}{1+\tanh ^{2}x}}\\\sinh(2x)&={\frac {2\tanh x}{1-\tanh ^{2}x}}\\\cosh(2x)&={\frac {1+\tanh ^{2}x}{1-\tanh ^{2}x}}\end{aligned}}

and the "half-argument formulas"^[19]

\sinh {\frac {x}{2}}={\sqrt {{\frac {1}{2}}(\cosh x-1)}}\,

Note: This is equivalent to its circular counterpart multiplied by −1.

\cosh {\frac {x}{2}}={\sqrt {{\frac {1}{2}}(\cosh x+1)}}\,

Note: This corresponds to its circular counterpart.

\tanh {\frac {x}{2}}={\sqrt {\frac {\cosh x-1}{\cosh x+1}}}={\frac {\sinh x}{\cosh x+1}}={\frac {\cosh x-1}{\sinh x}}=\coth x-\operatorname {csch} x.

\coth {\frac {x}{2}}=\coth x+\operatorname {csch} x.

The derivative of sinh x is cosh x and the derivative of cosh x is sinh x; this is similar to trigonometric functions, albeit the sign is different (i.e., the derivative of cos x is −sin x).

The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones 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

From the definitions of the hyperbolic sine and cosine, we can derive the following identities:

e^{x}=\cosh x+\sinh x

and

e^{-x}=\cosh x-\sinh x

These expressions are analogous to the expressions for sine and cosine, based on Euler's formula, as sums of complex exponentials.