Hyperbolic function

Standard analytic expressions

The hyperbolic functions are:

Hyperbolic sine:

Hyperbolic cosine:

Hyperbolic tangent:

Hyperbolic cotangent: x ≠ 0

Hyperbolic secant:

Hyperbolic cosecant: x ≠ 0

Hyperbolic functions can be introduced via imaginary circular angles:

Hyperbolic sine:

Hyperbolic cosine:

Hyperbolic tangent:

Hyperbolic cotangent:

Hyperbolic secant:

Hyperbolic cosecant:

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

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

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:

area = ∫ a b cosh ⁡ ( x )   d x = ∫ a b 1 + ( d d x cosh ⁡ ( x ) ) 2   d x = 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:

1 2 f ″ = f 3 − f ; f ( 0 ) = f ′ ( ∞ ) = 0

Useful relations

Odd and even functions:

sinh ⁡ ( − x ) = − sinh ⁡ x cosh ⁡ ( − x ) = cosh ⁡ x

Hence:

tanh ⁡ ( − x ) = − tanh ⁡ x coth ⁡ ( − x ) = − coth ⁡ x sech ⁡ ( − x ) = sech ⁡ x csch ⁡ ( − x ) = − csch ⁡ x

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

arsech ⁡ x = arcosh ⁡ 1 x arcsch ⁡ x = arsinh ⁡ 1 x arcoth ⁡ x = artanh ⁡ 1 x

Hyperbolic sine and cosine satisfy:

cosh ⁡ x + sinh ⁡ x = e x cosh ⁡ x − sinh ⁡ x = e − x cosh 2 ⁡ x − sinh 2 ⁡ x = 1

the last which is similar to the Pythagorean trigonometric identity.

One also has

sech 2 ⁡ x = 1 − tanh 2 ⁡ x csch 2 ⁡ x = coth 2 ⁡ x − 1

for the other functions.

Sums of arguments

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 ) = tanh ⁡ x + tanh ⁡ y 1 + tanh ⁡ x tanh ⁡ y

particularly

cosh ⁡ ( 2 x ) = sinh 2 ⁡ x + cosh 2 ⁡ x = 2 sinh 2 ⁡ x + 1 = 2 cosh 2 ⁡ x − 1 sinh ⁡ ( 2 x ) = 2 sinh ⁡ x cosh ⁡ x

Also:

sinh ⁡ x + sinh ⁡ y = 2 sinh ⁡ x + y 2 cosh ⁡ x − y 2 cosh ⁡ x + cosh ⁡ y = 2 cosh ⁡ x + y 2 cosh ⁡ x − y 2

Subtraction formulas

sinh ⁡ ( x − y ) = sinh ⁡ ( x ) cosh ⁡ ( y ) − cosh ⁡ ( x ) sinh ⁡ ( y ) cosh ⁡ ( x − y ) = cosh ⁡ ( x ) cosh ⁡ ( y ) − sinh ⁡ ( x ) sinh ⁡ ( y )

Also:

sinh ⁡ x − sinh ⁡ y = 2 cosh ⁡ x + y 2 sinh ⁡ x − y 2 cosh ⁡ x − cosh ⁡ y = 2 sinh ⁡ x + y 2 sinh ⁡ x − y 2

Source.

Half argument formulas

sinh ⁡ ( x 2 ) = sinh ⁡ ( x ) 2 ( cosh ⁡ ( x ) + 1 ) = sgn ⁡ ( x ) cosh ⁡ ( x ) − 1 2

where sgn is the sign function.

cosh ⁡ ( x 2 ) = cosh ⁡ ( x ) + 1 2 tanh ⁡ ( x 2 ) = sinh ⁡ ( x ) cosh ⁡ ( x ) + 1 = sgn ⁡ ( x ) cosh ⁡ ( x ) − 1 cosh ⁡ ( x ) + 1 = e x − 1 e x + 1

If x ≠ 0, then

tanh ⁡ ( x 2 ) = cosh ⁡ ( x ) − 1 sinh ⁡ ( x ) = coth ⁡ ( x ) − csch ⁡ ( x )

Inverse functions as logarithms

arsinh ⁡ ( x ) = ln ⁡ ( x + x 2 + 1 ) arcosh ⁡ ( x ) = ln ⁡ ( x + x 2 − 1 ) ; x ≥ 1 artanh ⁡ ( x ) = 1 2 ln ⁡ ( 1 + x 1 − x ) ; | x | < 1 arcoth ⁡ ( x ) = 1 2 ln ⁡ ( x + 1 x − 1 ) ; | x | > 1 arsech ⁡ ( x ) = ln ⁡ ( 1 x + 1 x 2 − 1 ) = ln ⁡ ( 1 + 1 − x 2 x ) ; 0 < x ≤ 1 arcsch ⁡ ( x ) = ln ⁡ ( 1 x + 1 x 2 + 1 ) = ln ⁡ ( 1 + x 2 + 1 x ) ; x ≠ 0

Derivatives

d d x sinh ⁡ x = cosh ⁡ x d d x cosh ⁡ x = sinh ⁡ x d d x tanh ⁡ x = 1 − tanh 2 ⁡ x = sech 2 ⁡ x = 1 / cosh 2 ⁡ x d d x coth ⁡ x = 1 − coth 2 ⁡ x = − csch 2 ⁡ x = − 1 / sinh 2 ⁡ x if x ≠ 0 d d x   sech x = − tanh ⁡ x   sech x d d x   csch x = − coth ⁡ x   csch x if x ≠ 0 d d x arsinh x = 1 x 2 + 1 d d x arcosh x = 1 x 2 − 1 if 1 < x d d x artanh x = 1 1 − x 2 if -1 < x < 1 d d x arcoth x = 1 1 − x 2 if x < -1 or 1 < x d d x arsech x = − 1 x 1 − x 2 if 0 < x < 1 d d x arcsch x = − 1 | x | 1 + x 2 if x ≠ 0

Second derivatives

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

d 2 d x 2 sinh ⁡ x = sinh ⁡ x 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 e x and e − x , and the zero function f ( x ) = 0 .

Standard integrals

∫ sinh ⁡ ( a x ) d x = a − 1 cosh ⁡ ( a x ) + C ∫ cosh ⁡ ( a x ) d x = a − 1 sinh ⁡ ( a x ) + C ∫ tanh ⁡ ( a x ) d x = a − 1 ln ⁡ ( cosh ⁡ ( a x ) ) + C ∫ coth ⁡ ( a x ) d x = a − 1 ln ⁡ ( sinh ⁡ ( a x ) ) + C ∫ sech ⁡ ( a x ) d x = a − 1 arctan ⁡ ( sinh ⁡ ( a x ) ) + C ∫ csch ⁡ ( a x ) d x = a − 1 ln ⁡ ( tanh ⁡ ( a x 2 ) ) + C = a − 1 ln ⁡ | csch ⁡ ( a x ) − coth ⁡ ( a x ) | + C

The following integrals can be proved using hyperbolic substitution: ∫ 1 a 2 + u 2 d u = arsinh ⁡ ( u a ) + C ∫ 1 u 2 − a 2 d u = arcosh ⁡ ( u a ) + C ∫ 1 a 2 − u 2 d u = a − 1 artanh ⁡ ( u a ) + C ; u 2 < a 2 ∫ 1 a 2 − u 2 d u = a − 1 arcoth ⁡ ( u a ) + C ; u 2 > a 2 ∫ 1 u a 2 − u 2 d u = − a − 1 arsech ⁡ ( u a ) + C ∫ 1 u a 2 + u 2 d u = − a − 1 arcsch ⁡ | u a | + C

where C is the constant of integration.

Taylor series expressions

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

sinh ⁡ x = x + x 3 3 ! + x 5 5 ! + x 7 7 ! + ⋯ = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 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 + x 2 2 ! + x 4 4 ! + x 6 6 ! + ⋯ = ∑ n = 0 ∞ x 2 n ( 2 n ) !

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.

tanh ⁡ x = x − x 3 3 + 2 x 5 15 − 17 x 7 315 + ⋯ = ∑ n = 1 ∞ 2 2 n ( 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! , | x | < π 2 coth ⁡ x = x − 1 + x 3 − x 3 45 + 2 x 5 945 + ⋯ = x − 1 + ∑ n = 1 ∞ 2 2 n B 2 n x 2 n − 1 ( 2 n ) ! , 0 < | x | < π sech x = 1 − x 2 2 + 5 x 4 24 − 61 x 6 720 + ⋯ = ∑ n = 0 ∞ E 2 n x 2 n ( 2 n ) ! , | x | < π 2 csch x = x − 1 − x 6 + 7 x 3 360 − 31 x 5 15120 + ⋯ = x − 1 + ∑ n = 1 ∞ 2 ( 1 − 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) ! , 0 < | x | < π

where:

B n is the nth Bernoulli number E n is the nth Euler number

Comparison with circular functions

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 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.

Mellon Haskell of University of California, Berkeley described the basis of hyperbolic functions in areas of hyperbolic sectors in an 1895 article in Bulletin of the American Mathematical Society (see External links). He refers to the hyperbolic angle as an invariant measure with respect to the squeeze mapping just as circular angle is invariant under rotation.

Identities

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule 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

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 ) = tanh ⁡ ( x ) + tanh ⁡ ( y ) 1 + tanh ⁡ ( x ) tanh ⁡ ( y )

the "double argument formulas"

sinh ⁡ ( 2 x ) = 2 sinh ⁡ x cosh ⁡ x cosh ⁡ ( 2 x ) = cosh 2 ⁡ x + sinh 2 ⁡ x = 2 cosh 2 ⁡ x − 1 = 2 sinh 2 ⁡ x + 1 tanh ⁡ ( 2 x ) = 2 tanh ⁡ x 1 + tanh 2 ⁡ x sinh ⁡ ( 2 x ) = 2 tanh ⁡ x 1 − tanh 2 ⁡ x cosh ⁡ ( 2 x ) = 1 + tanh 2 ⁡ x 1 − tanh 2 ⁡ x

and the "half-argument formulas"

sinh ⁡ x 2 = 1 2 ( cosh ⁡ x − 1 )    Note: This is equivalent to its circular counterpart multiplied by −1. cosh ⁡ x 2 = 1 2 ( cosh ⁡ x + 1 )    Note: This corresponds to its circular counterpart. tanh ⁡ x 2 = cosh ⁡ x − 1 cosh ⁡ x + 1 = sinh ⁡ x cosh ⁡ x + 1 = cosh ⁡ x − 1 sinh ⁡ x = coth ⁡ x − csch ⁡ x . coth ⁡ x 2 = coth ⁡ x + 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.

Hyperbolic functions for complex numbers

Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The functions sinh z and cosh z are then holomorphic.

Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:

e i x = cos ⁡ x + i sin ⁡ x e − i x = cos ⁡ x − i sin ⁡ x

so:

cosh ⁡ ( i x ) = 1 2 ( e i x + e − i x ) = cos ⁡ x sinh ⁡ ( i x ) = 1 2 ( e i x − e − i x ) = i sin ⁡ x cosh ⁡ ( x + i y ) = cosh ⁡ ( x ) cos ⁡ ( y ) + i sinh ⁡ ( x ) sin ⁡ ( y ) sinh ⁡ ( x + i y ) = sinh ⁡ ( x ) cos ⁡ ( y ) + i cosh ⁡ ( x ) sin ⁡ ( y ) tanh ⁡ ( i x ) = i tan ⁡ x cosh ⁡ x = cos ⁡ ( i x ) sinh ⁡ x = − i sin ⁡ ( i x ) tanh ⁡ x = − i tan ⁡ ( i x )

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period 2 π i ( π i for hyperbolic tangent and cotangent).