Hyperbolic Trigonometric Functions

the Exponential Function, a review

Periodicity is a characteristic feature of Elliptic Functions. Elliptic Functions may be either doubly-periodic, singly-periodic, or (triviallly) non-periodic. The Exponential Function is the singly-periodic Elliptic Function. Thus, the Exponential Function possesses the modulus of periodicity of 2 pi i

exp(x + 2 pi i) = exp(x) for any x in Complex.

And, by saying that 2 pi i is the "modulus of periodicity" we mean that 2 pi i is the smallest, in magnitude, constant for which the foregoing identity is true. Incidentally, we thus have defined pi. An approximate value of pi is 3.14159, a Real number. Each of the trigonometric funcions will inherit this modulus of periodicity, except that the tangent and cotanget functions have a modulus of periodicity only half the size.

Possesion of an addition theorem is another characteristic feature of Elliptic Functions. The addition theorem for the Exponential Function is

exp(x + y) = exp(x) exp(y) for any (x, y) in the Cartesian product of Complex by Complex.

The third characteristic property of the Elliptic Functions is that they satisfy a certain differential equation. Thus, the exponential function may be characterized by the differential equation

d exp(x) / dx - exp(x) = 0, with exp(0) = 1,

for any x in Complex.

The McLaurin's series for the exponential function is

exp(x) = 1 + x + x^2 / 2 + x^3 / 3 + ... + x^n / n! + ....

Hyperbolic Trigonometric Functions

Definitions

For an x in Complex, we define the nine Hyperbolic Trigonometric functins as follows:

The sine sinh(x) = (exp(x) - exp(- x)) / 2
The cosine cosh(x) = (exp(x) + exp(- x)) / 2
The tangent tanh(x) = sinh(x) / cosh(x)
The cotangent coth(x) = 1 / tanh(x)
The secent sech(x) = 1 / cosh(x)
The cosecent csch(x) = 1 / sinh(x)
The versed sine versinh(x) = 1 - cosh(x)
The coversed sine does not translate usefully
The haversed sine haversin(x) = (1 - cosh(x)) / 2

Identities

For any x in the Cartesian product of Complex by Complex, from the foregoing, the following identities are obvious:

(cosh(x))^2 = 1 + (sinh(x))^2 hint: substitute the definitions.
1 = (sech(x))^2 + (tanh(x))^2 hint: multiply the foregoing by (sech(x))^2.
(coth(x))^2 = (csch(x))^2 + 1 hint: multiply the first of the two foregoing by (csch(x))^2.

Addition Theorems

Real

For any (x, y) in the Cartesian product of Complex by Complex, from the definitions of the Hyperbolic Trigonometric function, the following real addition theorems are obvious:

sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y) hint: substitute the definitions.
cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y) hint: substitute the definitions.
tanh(x + y) = (tanh(x) + tanh(y)) / (1 + tanh(x) tanh(y)) hint: divide the preceding two equations.
coth(x + y) = (1 + coth(x) coth(y)) / ( coth(x) + coth(y)) hint: find the reiprocal of the foregoing.
The addition theorems for the hyperbolic secent, cosecent, versed sine, and haversed sine are not interesting.

Products

By substitution of the addition formulae into the right-hand side of the following, one may obtain these real product theorems:

sinh(x) cosh(y) = (sinh(x + y) + sinh(x - y)) / 2
cosh(x) cosh(y) = (cosh(x + y) + cosh(x - y)) / 2
sinh(x) sinh(y) = (cosh(x + y) - cosh(x - y)) / 2

Sums or Differences

Let u = x + y and v = x - y. Substitution in the foregoing three equations yields the Sums or Differencess

sinh(u) + sinh(v) = 2 sin(h(u + v) / 2) cosh((u - v) / 2)
cosh(u) + cosh(v) = 2 cosh(u + v) / 2) cosh(u - v) / 2)
cosh(u) - cosh(v) = 2 sin(h(u + v) / 2) sinh((u - v) / 2)

Complex

For any (x, y) in the Cartesian product of Complex by Complex, from the foregoing read addifion theorems, with the replacement of y by i y and the use of the circular trigonometric functions, the following complex addition theorems are obvious

exp(x + i y) = exp(x) (cos(y) + i sin(y)).
sinh(x + i y) = sinh(x) cos(y) + i cosh(x) sin(y).
cosh(x + i y) = cosh(x) cos(y) + i sinh(x) sin(y).
tanh(x + i y) = (tanh(x) + i tan(y)) / (1 + i tanh(x) tan(y)).
coth(x + i y) = (coth(x) cot(y) + i) / (i coth(x) + cot(y)).
The complex addition theorems for the hyperbolic secent, cosecent, versed sine, and haversed sine are not interesting.

The special cases where x is zero are as follows

exp(i y) = cos(y) + i sin(y).
sinh(i y) = i sin(y).
cosh(i y) = cos(y).
tanh(i y) = i tan(y).
coth(i y) = - i cot(y).
sech(i y) = sec(y).
csch(i y) = - i csc(y).
versinh(i y) = versin(y).
haversinh(i y) = haversin(y).

On the other hand, we may invert the first three of the foregoing complex addition theorems. Set the right-hand side equal to u + i v. Then collect the real and imaginary parts on one side of the equation; each part has to be zero. Then by employing the identities, we obtain

Arcexp(u + i v) = ln(u + i v) = (1 / 2) ln(u^2 + v^2) + i Arctan(u / v)
Arcsinh(u + i v) = ???
Arccosh(u + i v) = ???

Double-Angle Formulae

In the foregoing real addition theorems, take y equal to x, to yield the following double-angle formulae:

sinh(2 x) =2 sinh(x) cosh(x).
cosh(2 x) = (cosh(x))^2 + (sinh(x))^2.
tanh(2 x) = 2 tanh(x) / (1 + (tanh(x))^2).
coth(2 x) = (1 + (coth(x))^2) / (2 coth(x)).
sech(2 x) = (sech(x) csch(x))^2 / ((sech(x))^2 + (csch(x))^2).
csch(2 x) = (sech(x) csch(x))^2 / (2 sech(x) csch(x)).
versinh(2 x) = - 2 (sinh(x))^2).
haversinh(2 x) = - (sinh(x))^2).

By the use of the first of the identities, the double angle formula for the hyperbolilc cosine may be written in either of the additinal two forms:

cosh(2 x) = 1 + 2 (sinh(x))^2.
cosh(2 x) = 2 (cosh(x))^2 - 1.

Half-Angle Formulae

The half-angle formulae are as follows:

sinh(x / 2) = +- sqrt(cosh(x) - 1) / 2 hint: solve the first of the two immediately preceding double-angle equations.
cosh(x / 2) = +- sqrt(cosh(x) + 1) / 2 hint: solve the second of the two immediately preceding double-angle equations.
tanh(x / 2) =
- = +- sqrt((cosh(x) - 1) / (cosh(x) + 1)) hint: diivide the foregoing.
- = sinh(x) / (cosh(x) + 1) hint: multiply both the numerator and the denominator of the first of the formulae for the half-angle of the hyperbolic-tangent.by (cosh(x) + 1) and employ the identity involving the hyperbolic sine and cosine functions.
- = (cosh(x) - 1) / sinh(x) hint: as in the foregoing, but by (cosh(x) - 1).
The half-angle formulae for the hyperbolic cotangent, secent, cosecent, versed sine, and haversed sine are not interesting.

Inverse Hyperbolic in Terms of Logarithms

Let y = Arcsinh(x) and solve for x to obtain

x = sinh(y) = (exp(y) - exp(- y)) / 2 = (exp(2 y) - 1) / exp(y).

Solution of this quadratic equation yields the forst of the following formulae of the inverse hyperboic trigonometric functions in terms of the logarithmic function.

Arcsinh(x) = ln(x + sqrt(x^2 + 1))
Arccosh(x) = ln(x + sqrt(x^2 - 1))
Arctanh(x) = (1 / 2) ln((1 + x) / (1 - x))
Arccoth(x) = (1 / 2) ln((x + 1) / (x - 1))
Arcsech(x) = ln((1 + sqrt(1 - x^2)) / x)
Arccsch(x) = ln((1 + sqrt(1 + x^2)) / x)

Hyperbola

A parametric equation of a hyperbola, in the Cartesian product of Complex by Complex, is given by

(x, y) = (a cosh(t), b sinh(t)) for any t in Complex and any constant (a, b), called the semi-axes, in the Cartesian product of Complex by Complex. In the case of real t, this formula yields only one branch of the hyperbola. To obtain the other branch, change the sign of a.

Passive (that is with time being unknown) navigation employs hyperbolas and hyperboloids of revolution of two-sheets.

Historically, these functions have been called hyperbolic because of this parametrization of a nyperbola.

Calculus

Derivatives

These derivative formulae are obvious from the definition of the functions. For any x in Complex we have the following derivative formulae:

Direct

d exp(x) / dx = exp(x).
d sinh(x) / dx = cosh(x).
d cosh(x) / dx = sinh(x).
d tanh(x) / dx = (sech(x))^2.
d coth(x) / dx = - (csch(x))^2.
d sech(x) / dx = - tanh(x) sech(x).
d csch(x) / dx = - coth(x) csch(x).
d versinh(x) / dx = - sinh(x).
d haversinh(x) / dx = - sinh(x) / 2.

Inverse

Let x = sinh(y) and differentiate it to obtain dx / dy = cosh(y). Employ the appropriateidentities to obtain dx / dy = sqrt(1 + (sinh(y))^2). Then dy / dx = 1 / sqrt( + (sihn(y))^2). Thus, we have obtained the first of the derivative formulae of the inverse hyperbolic trigonometric functions

d Arcsinh(x) / dx = 1 / sqrt(x^2 + 1)
d Arccosh((x) / dx = 1 / sqrt(x^2 - 1)
d Arctanh(x) / dx = 1 / (1 - x^2)
d Arccoth(x) / dx = 1 / (x^2 - 1)
d Arcsech(x) / dx = - x sqrt(1 - x^2)
d Arccsch(x) / dx = - x sqrt(1 + x^2))

Their primary utility is as antiderivatives.

Integrals

The Riemann Integral is an ant-derivative, thus each of these integral formulae may be verified by differentiation. For any x in Complex we have the integral formuale:

int(sinh(x)) = cosh(x) + C.
int(cosh(x)) = sinh(x) + C.
int(tanh(x)) = ln(cosh(x)) + C .
int(coth(x)) = - ln(sinh(x)) + C.
int(sech(x)) = - ln(sec(x) + tan(x)) + C.
int(csch(x)) = - ln(csc(x) + cot(x)) + C.
int(versin(x)) = x + cos(x) + C.
int(coversin(x)) = x - sin(x) + C.
int(haversin(x)) = (x + cos(x)) / 2 + C.

Infinite Expansions

McLaurin's Series

The McLaurin's series for the hyperbolic sine and cosine may be obtained from their definitions and the series for the exponential function.

Take the infinite Geometric series

1 / (1 - x) = 1 + x + x^2 + x^3 + ... + x^n + ....

for any x, in Complex, whose the absolute value is less than one. Replace x by x^2 and integrate to obtain the McLaurin's series for the hyperbolic arctangent.

sine sinh(x) = x + x^3 / 6+ x^5 / 120 + ... + x^(2 n + 1) / (2 n + 1)! + ....
cosine cosh(x) = 1 + x^2 / 2 +x^4 / 24 + ... + x^(2 n) / (2 n)! + ....
The McLaurin's series formulae for the hyperbolic tangent, cotangent, secent, cosecent, versed sine, coverssed sine, and haversed sine are not interesting.
arctangent arctanh(x) = x + x^2 / 2 + x^3 / 3 + ... + x^(n + 1) / (n + 1) + .... provided that abs(x) < 1.
The McLaurin's series formulae for the hyperbolic arc sine, arc cosine, arc cotangent, arc secent, arc cosecent, arc versed sine, arc coverssed sine, and arc haversed sine are not interesting. The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the identities and definitions.

Infinite Products

The hyperbolic infinite products are not interesting. They have to be obtained from the corresponding circular infinite products, employing the purely imaginary relationship between the corresponding functions.

Circular Trigonometric Functions

The purely imaginary counterpart of the Circular Trigonometric functions is called the Circular Trigonometric functions.